A Macro-Finance Approach to Sovereign Debt Spreads and Returns * Fabrice Tourre † University of Chicago [Link to latest draft and online appendix] January 29, 2017 Abstract Foreign currency sovereign bond spreads tend to be higher than historical sovereign credit losses, and cross-country spread correlations are larger than their macro-economic counterparts. Foreign currency sovereign debt exhibits positive and time-varying risk premia, and standard linear asset pricing models using US-based factors cannot be rejected. The term structure of sovereign credit spreads is upward sloping, and inverts when either (a) the country’s fundamentals are bad or (b) measures of US equity or credit market stress are high. I develop a quantitative and tractable continuous-time model of endogenous sovereign default in order to account for these stylized facts. My framework leads to semi-closed form expressions for certain key macro-economic and asset pricing moments of interest, helping disentangle which of the model features influences credit spreads, expected returns and cross-country correlations. Standard pricing kernels used to explain properties of US equity returns can be nested into my quantitative framework in order to test the hypothesis that US-based bond investors are marginal in sovereign debt markets. I show how to leverage my model to study the early 1980’s Latin American debt crisis, during which high short term US interest rates and floating rate dollar-denominated debt led to a wave of sovereign defaults. * First draft January 2016. I would like to thank my advisors Fernando Alvarez (Chair), Lars Hansen, Zhiguo He and Rob Shimer for their continuous support, guidance and encouragements. I would also like to thank Gabriela Antonie, Simcha Barkai, George Constantinides, Paymon Khorrami, Arvind Krishnamurthy, Konstantin Milbradt, Lubos Pastor, Mark Wright, and the participants of the Economic Dynamics and Capital Theory workshops for their comments and suggestions. I also would like to gratefully acknowledge financial support from the Macro-Finance Modeling Group as well as the Stevanovich Center. The views in this paper are solely mine. Any and all mistakes in this paper are mine as well. † Fabrice Tourre: PhD Candidate, Department of Economics, University of Chicago, 1126 E. 59th Street – Saieh Hall – Chicago, IL 60637. Email: [email protected]
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A Macro-Finance Approach toSovereign Debt Spreads and Returns∗
Fabrice Tourre†
University of Chicago
[Link to latest draft and online appendix]
January 29, 2017
Abstract
Foreign currency sovereign bond spreads tend to be higher than historical sovereign
credit losses, and cross-country spread correlations are larger than their macro-economic
counterparts. Foreign currency sovereign debt exhibits positive and time-varying risk
premia, and standard linear asset pricing models using US-based factors cannot be
rejected. The term structure of sovereign credit spreads is upward sloping, and inverts
when either (a) the country’s fundamentals are bad or (b) measures of US equity or
credit market stress are high. I develop a quantitative and tractable continuous-time
model of endogenous sovereign default in order to account for these stylized facts.
My framework leads to semi-closed form expressions for certain key macro-economic
and asset pricing moments of interest, helping disentangle which of the model features
influences credit spreads, expected returns and cross-country correlations. Standard
pricing kernels used to explain properties of US equity returns can be nested into my
quantitative framework in order to test the hypothesis that US-based bond investors
are marginal in sovereign debt markets. I show how to leverage my model to study the
early 1980’s Latin American debt crisis, during which high short term US interest rates
and floating rate dollar-denominated debt led to a wave of sovereign defaults.
∗First draft January 2016. I would like to thank my advisors Fernando Alvarez (Chair), Lars Hansen,Zhiguo He and Rob Shimer for their continuous support, guidance and encouragements. I would also like tothank Gabriela Antonie, Simcha Barkai, George Constantinides, Paymon Khorrami, Arvind Krishnamurthy,Konstantin Milbradt, Lubos Pastor, Mark Wright, and the participants of the Economic Dynamics andCapital Theory workshops for their comments and suggestions. I also would like to gratefully acknowledgefinancial support from the Macro-Finance Modeling Group as well as the Stevanovich Center. The views inthis paper are solely mine. Any and all mistakes in this paper are mine as well.†Fabrice Tourre: PhD Candidate, Department of Economics, University of Chicago, 1126 E. 59th Street
Driven by low real interest rates, high commodity prices and easy credit, Latin American
external debt grew significantly in the 1970s. The Volcker shock, combined with debt con-
tracts indexed to US short term rates, contributed to the subsequent debt crisis and the “lost
decade” suffered by many Latin American countries in the 1980s. A quarter of a century
later, in the fall 2008, the US subprime crisis morphed into a global financial crisis, leading
to a shut down of emerging economies’ access to international credit markets and a violent
widening of their sovereign spreads. Those two episodes highlight the central importance
of the supply of capital for sovereign debt dynamics. However, a large component of the
international macroeconomic literature on sovereign credit risk uses economic models where
external creditors are risk-neutral, assuming away any possible link between investors’ at-
tributes and government financing and default decisions1. The modeling hypothesis of this
line of research stems from its main focus on macroeconomic quantities (such as the current
account balance and the debt-to-GDP ratio) as opposed to prices, and from the difficulty of
adding one or several dimensions to already complex models of endogenous default. Sepa-
rately, the fixed income asset pricing literature on sovereign debt takes seriously investors’
risk attributes when explaining properties of sovereign credit spreads and returns, but it does
so at the expense of modeling the underlying asset cash-flows and their dynamic properties.
Indeed, its primary objective is to use bond and credit derivatives’ market prices in order to
estimate hazard rate of default processes, without having the need to relate them to economic
fundamentals.
My paper bridges the gap between these two seemingly disconnected literatures by offering
a new model of endogenous sovereign default where the supply of capital takes on a prominent
role, as supported by known stylized facts as well as new evidence I document in my empirical
work. Thanks to its reduced dimensionality, the proposed framework remains tractable and
allows me to obtain semi-closed form expressions for several macroeconomic and asset pricing
moments of interest, helping disentangle which features of the model are essential to generate
specific moments of the data. In addition, it facilitates the estimation and testing of the
model, and an in-depth analysis of the government financing and default policies. It can then
be used to answer numerous questions: how much of sovereign governments’ financing costs
can be attributed to bond investors’ risk characteristics, and how much to country-specific
macroeconomic risks? Are sovereign debt return co-movements mostly due to correlated
fundamentals, or the fact that a common bond buyer base is marginal in sovereign bond
markets? Can supply-side shocks to capital markets rationalize the magnitude of current
1Two notable exceptions are Borri and Verdelhan (2011) and Lizarazo (2013).
1
account reversals observed in the context of “sudden-stops” suffered by emerging market
economies in Latin America in the early 1980s, or in South East Asia in the late 1990s?
In the empirical section of my paper, I infer market-implied (sometimes called “risk-
neutral”) default intensities from sovereign credit-default swap (“CDS”) premia, and then
compute returns on CDS contracts. Leveraging my constructed data-set, I document three
sets of empirical facts that are the counterparts to known properties of foreign currency
sovereign bond prices and returns. Those facts will not only guide the construction of my
model but also will be used for estimation and testing.
First, I provide evidence that investors in sovereign debt markets do not behave risk-
neutrally. To do so, I show that market-implied default intensities are significantly larger
than historical default frequencies, and that sovereign CDS’ expected excess returns are
positive. Together, these empirical properties of sovereign debt spreads and returns illustrate
the two sides of the same coin: creditors require compensation for being exposed to a risk
(the sovereign default risk) that co-moves with their pricing kernel. While these stylized
facts have already been investigated by Broner, Lorenzoni, and Schmukler (2013) and Borri
and Verdelhan (2011) in the context of foreign currency sovereign bonds, I contribute to the
empirical debate by showing that this property of sovereign credit prices and returns also
holds for CDS contracts.
Second, the data supports not only that sovereign debt investors are risk-averse, but also
that their pricing of risk is time-varying and relates to measures of US credit and equity
market risks. Indeed, the difference between market-implied and historical default intensities
is time-varying and cannot be explained by time-varying country-specific macroeconomic risk
factors. This stylized fact has been documented previously by Longstaff et al. (2011) and
Pan and Singleton (2008), who analyzed local and global factors that explain movements in
sovereign CDS premia. Using my constructed CDS return data, I then perform standard
linear asset pricing tests, using US equity market returns, and I fail to reject the hypothesis
that a linear stochastic discount factor can price my set of excess returns. This exercise lends
support to the analysis performed by Borri and Verdelhan (2011) in the context of sovereign
bond returns. Finally, I show that cross-country CDS return correlations are significantly
larger than their macroeconomic counterparts, suggesting that a common bond buyer base
is marginal in foreign currency sovereign debt markets.
While these facts, taken together, help us understand the required characteristics of
a sovereign investors’ pricing kernel, they are silent on the type of mechanism leading to
sovereign defaults, and how supply side factors may impact a sovereign government’s bor-
rowing and default decisions. I speak to this question by illustrating a third set of facts,
related to the term structure of market-implied default intensities and returns. First, I show
2
that the term structure of default intensities is upward sloping for most countries, but it
flattens and inverts if either (i) a country’s fundamentals deteriorate, or (ii) measures of US
credit or equity market stress are high. Second, I show that holding period excess returns
are increasing with the maturity of the CDS contract – this latter fact being documented
by Broner, Lorenzoni, and Schmukler (2013) in the context of foreign currency sovereign
bonds. Both properties of the term structure of spreads and returns are consistent with a
“first hitting time” model, where a sovereign default is triggered by some – possibly endoge-
nous – mean-reverting fundamental variable exceeding a certain threshold that depends on
aggregate financial market conditions.
What might this macroeconomic “fundamental” variable be? In my theoretical setup,
it is the debt-to-GDP ratio. I leverage the canonical sovereign default model of Eaton and
Gersovitz (1981), further enhanced by Arellano (2008) and Aguiar and Gopinath (2006), and
develop a quantitative continuous time model of sovereign debt issuances and defaults, in
which a government uses non-state contingent debt sold to foreign creditors for the purpose
of consumption smoothing and consumption tilting2. The government’s inability to commit to
repay its debt leads to default risk. Following a default, the country suffers an instantaneous
discrete drop in output and loses access to capital markets for an exponentially distributed
time period. Using a modeling device used in Nuno and Thomas (2015), the country then re-
enters financial markets with a lower debt burden, the result of an un-modeled renegotiation
with its creditors. The sovereign debt-to-GDP ratio naturally arises as the fundamental state
variable – a consequence of the homotheticity of the government’s objective function and the
linearity of output and debt dynamics. I deviate from the canonical sovereign debt models
along several dimensions. In order to capture realistic features of debt contracts used by
governments in international capital markets, I model a sovereign issuing long-term debt, as
opposed to short-term debt3. For tractability, those bonds have an exponential amortization
schedule, a common tool in the corporate credit literature (Leland (1998) was the first paper
to my knowledge to use this modeling assumption), and recently adopted by the sovereign
default literature (Hatchondo and Martinez (2009), Chatterjee and Eyigungor (2015) for
example). Since my focus is on the supply of capital and its impact on sovereign bond prices
and returns, I introduce investors, whose preferences and equilibrium consumption lead to a
pricing kernel that features regime-dependent risk free rates and risk prices, in the spirit of
Chen (2010). Those regimes act as a second – exogenous and discrete – state variable that
describes the international capital market environment.
2As is typically the case in the international macroeconomic literature, the sovereign government will bemore impatient than its creditors, providing an incentive to borrow in order to consume early.
3Since my model is cast in continuous-time, infinitesimally short term debt would not carry any defaultrisk, as showed more formally in Section A.1.1.
3
My modeling ingredients lead to sovereign spreads that are greater than model-implied
historical credit losses, as I document in the empirical part of the paper. For a panel of emerg-
ing market countries, I can then estimate the proportion of the average credit spread that
can be attributed to (a) pure default risk and (b) the risk premium charged by international
investors. I find that approximately 30% of sovereign governments’ financing costs (over and
above the risk-free rate) is attributable to required compensation paid to investors for taking
on risks that are correlated with their marginal utilities. In the model, spread volatilities
stem not only from output shocks, but also from stochastic discount factor (“SDF”) shocks,
and are thus close to spread volatilities in the data, a moment notoriously difficult to match
with standard models (Aguiar et al. (2016b)). For the same reason, cross-country sovereign
spread correlations are larger than cross-country output correlations. By turning on and
off those SDF shocks, I can then infer the proportion of such cross-country spread correla-
tion that relates to correlated fundamentals, and the proportion that relates to pricing by a
common stochastic discount factor.
In my model, the sovereign default decision features an optimal debt-to-GDP default
boundary that depends on the specific pricing kernel regime. Consistent with the data, this
characteristic of my model leads to upward sloping term structures of spreads and default
intensities for countries whose economic fundamentals are not too bad and in environments
where risk-prices are not too high. Transitions from a “good regime” (where prices of risk
are low for example) to a “bad regime” (with higher prices of risk) might cause the sovereign
to “jump-to-default”. Even if the sovereign government does not jump to default, it adjusts
downwards its financing policy, switching from running a current account deficit to a current
account surplus, and endogenously creating a sudden stop. For most of my countries of
focus, a jump from the most benign capital market environment to the worst environment
leads to current account adjustments of 3% to 5% of GDP, potentially explaining up to
half the adjustments observed in the data for the 1980s’ Latin American debt crisis or the
1997 Asian tiger crisis. SDF regime transitions are also associated with inversions of the
term structure of credit spreads, another feature of the data. When looking across multiple
countries, transitions from “good regimes” to “bad regimes” lead to sudden increases in
sovereign spreads as well as correlated defaults, arguably a feature of several sovereign debt
crisis. The jump-to-default risk induced by SDF shocks also leads to high short term credit
spreads, another stylized fact I document in the empirical section of my paper.
The continuous time framework I use has several key advantages over discrete time models
that have been the workhorse of the sovereign default literature. First and foremost, it
allows me to characterize fully the equilibrium of my model in the particular case where
the government is risk-neutral. I provide closed-form solutions for the country’s welfare, the
4
debt price, the optimal default boundary of the government, and compute the magnitude
of the current account reversal incurred upon an increase in the risk free rate or the price
of risk. Outside the knife-edge risk-neutral case, the continuous time framework facilitates
the transition from physical probabilities (under which the government optimizes) to risk-
neutral probabilities (under which creditors price the debt issued). It allows for semi-closed
form expressions of macro and asset pricing moments of interest, providing greater insight into
the specific impact of the model assumptions on endogenous quantities of focus. My model
features only two state variables – the debt-to-GDP ratio of the country being considered (a
continuous variable), and the SDF regime (a discrete variable). This low dimensionality of
the state space makes the framework more tractable than alternative models that have been
studied in the literature4. It permits an estimation of the key parameters of the model using
a panel of countries, and gives me the ability to test whether pricing kernels used to explain
properties of US equity returns can also explain properties of emerging market sovereign
bond returns. In my numerical applications, I test the pricing kernel featured in Lettau and
Wachter (2007) and show that that the level of risk-prices implied by such SDF is too low to
fully account for the expected excess return observed in the data for many emerging market
economies.
I finally highlight the flexibility of my framework by testing two new ideas. First, I
focus on the contractual structure of sovereign debt and study the spill-over effects of US
monetary policy on a government that issues debt whose coupon rate is indexed to US short-
term rates. While foreign currency sovereign bonds are nowadays mainly issued in fixed rate
form, Latin American countries used floating rate debt in the 1970’s and early 80’s, since the
funding came in the form of loans from US commercial banks. Given that my model features
time-varying risk-free rates, I can investigate the impact of US monetary policy on sovereign
default risk. In this paper, I show that a simple mechanism may have been at play both (a)
in the late 1970s, as Latin American economies took advantage of low US short term rates
to significantly increase their external sovereign debt and run current account deficits, and
(b) in the early 1980s’, as the US monetary authorities increased short term rates to fight
domestic inflation, increasing the debt servicing costs for Latin American governments and
ultimately triggering the defaults of Mexico and multiple other sovereign issuers after 1982.
In my model, in a low US short rate environment, floating rate sovereign issuers run current
4Other articles focused on sovereign spreads and returns include Borri and Verdelhan (2011), which feature4 state variables, and Aguiar et al. (2016b), which feature 5 state variables; in order to find an equilibriumin such models, not only does the researcher have to find a global solution to the value function of thegovernment (a function of all the state variables), but he also has to find the bond price schedule, whichdepends on both (i) the state variables and (ii) the amount of bonds that the government considers issuing.As will be clear in this paper, in continuous time the bond price schedule is no longer a function of theamount of bonds issued “in the next period”.
5
account deficits. When short term interest rates increase, a combination of lower debt prices
and a higher marginal cost of debt issuances make governments adjust their current account
balance by up to 15%, consistent in magnitude with what was observed empirically in 1982
in Mexico and other Latin American economies.
In a second application, I no longer assume a small open endowment economy but instead
introduce a simple “A-K” production technology with investment adjustment costs and capi-
tal quality shocks, as in Brunnermeier and Sannikov (2014). Sovereign debt is not only useful
for consumption smoothing and consumption tilting, but also to build the domestic capital
stock via investments. Thanks to the flexibility of my framework, the state space remains un-
changed, with only one additional control variable – investments – added for the small open
economy. In this modified environment, I show that two separate sources of debt overhang
can lead to under-investments: (a) after a sequence of bad capital quality shocks suffered in
the country’s production sector, or (b) after an SDF regime change from a mild capital mar-
ket environment to one with higher risk-prices. This enhanced model thus leads to a negative
correlation between sovereign spreads and investments, as observed in the data by Neumeyer
and Perri (2005) or Uribe and Yue (2006). It also provides a simple micro-foundation for
the output dynamics used in Aguiar and Gopinath (2004), Aguiar and Gopinath (2006) and
many other articles in the quantitative sovereign default literature, where log-output growth
is a mean-reverting variable. Finally, the debt overhang channel leads to an amplification
of the capital quality shocks, and thus to more volatile credit spreads and a wider ergodic
debt-to-GDP distribution, getting this class of models closer to the data.
This paper is organized as follows. The first part of the paper focuses on some empirical
facts of sovereign CDS premia and returns. I then develop a continuous time version of the
canonical model of sovereign borrowing and default, and enhance it by introducing a Markov
switching model of the stochastic discount factor used to price sovereign bonds. I estimate
the model and perform a variety of exercises to illustrate the tractability of the framework.
2 Stylized Facts
In this section, I summarize key stylized facts on foreign currency sovereign credit spreads
and returns. Many of these empirical observations have been highlighted in the past in con-
nection with research focused on foreign currency sovereign bonds. In the online appendix5, I
revisit those facts by looking at a different set of credit instruments: credit default swap con-
tracts referencing emerging market sovereign governments. My empirical analysis supports
the existing evidence on sovereign credit spreads and returns, and adds new observations
5See online appendix link at http://fabricetourre.com/research
missed by previous studies. I will use my empirical work to guide my model estimation and
validation.
(1) Hard currency sovereign credit spreads are higher than historical credit losses. This
fact is inconsistent with an assumption of investors’ risk-neutrality. It also means that
holding-period expected excess returns on foreign currency sovereign debt are positive.
This aspect of the data is highlighted by multiple studies, including for example Borri
and Verdelhan (2011) and Aguiar et al. (2016a). I add supporting evidence in my online
appendix, by showing that (i) hazard rates of default implied by the price of CDS con-
tracts are materially higher than historical default rates, and (ii) CDS expected excess
returns are positive.
(2) The differential between sovereign credit spreads and conditional expected credit losses
is time-varying, and is positively correlated with measures of US credit or equity market
risk. This fact is highlighted by Aguiar et al. (2016a) for example, who regress the level
of sovereign bond spreads onto the VIX index. It is also tightly related to a second
observation: holding-period excess returns on sovereign bonds are higher for countries
with higher US equity market beta. Borri and Verdelhan (2011) document this fact by
looking at returns on sovereign bonds in the EMBI index, and running standard cross-
sectional and time-series tests of the CAPM. In the online appendix, I obtain similar
results by using CDS returns as opposed to sovereign bond returns. I also emphasize
that CDS provide a “cleaner” measure of expected excess returns earned on sovereign
credit exposures than bonds – the latter not only being exposed to sovereign credit
risk, but also to the term structure of US interest rates6. A third observation, made for
example by Longstaff et al. (2007), is tightly connected to the other two: there is a strong
factor structure in the level of sovereign spreads, as supported by a principal component
analysis of the time series of CDS premia for multiple countries. In addition, the first
principal component in this decomposition is highly correlated with US equity market
returns. These three observations suggests the presence of US-based marginal investors
in foreign currency sovereign credit markets.
(3) Short term market-implied hazard rates of defaults are non-zero, leading to a rejection
of any model under which, at least at short horizons, defaults can be ruled out in some
6Most foreign currency sovereign bonds issued by small open economies nowadays are fixed rate bondsdenominated in USD. Researchers looking at time-series data on sovereign bonds rely on the EMBI index,compiled by JPMorgan, which provides, on a daily basis, an average price and average spread for a basket ofeligible obligations issued by each country included in the index. JPMorgan unfortunately does not providesecurity-specific prices, making it difficult to extract excess returns attributable purely to sovereign defaultrisk.
7
portions of the state space. An example of such model is one where default occurs
exclusively when a continuous process hits a barrier – a so-called “first-hitting-time”
model. This fact appears to be new in the sovereign default literature, and echos a similar
observation made by the corporate finance literature in connection with corporate credit
spreads.
(4) In time series, sovereign credit spreads are (a) negatively related to GDP growth, (b)
positively related to debt-to-GDP ratios, and (c) negatively related to measures of US
credit or equity market risk. (a) is documented by Neumeyer and Perri (2005) or Uribe
and Yue (2006), who however highlight that the relationship is weak. (b) is highlighted
in several studies, including Aguiar et al. (2016b). I provide new empirical evidence
supporting (b) and (c) by looking at CDS contracts.
(5) The term structure of sovereign credit spreads is upward-sloping, except for countries
whose credit spreads are high, for which the term structure is either flat or downward
sloping. The upward sloping term structure of spreads is highlighted by Pan and Single-
ton (2008), who focus on CDS contracts referencing Mexico, Turkey and South Korea.
The flattening and potential inversion of the term structure of credit spreads is briefly
noticed in Broner, Lorenzoni, and Schmukler (2013) and Arellano and Ramanarayanan
(2012). I provide additional supporting evidence for this feature of the data by focusing
on CDS for a panel of 27 emerging market economies. This fact is consistent with a
“first-hitting-time” model of sovereign default.
(6) The term structure of sovereign credit spreads “flattens” at times when international
investors’ risk prices are high. This feature of the data is different from fact (5), which
relates movements of the slope of spreads to the level of spreads for a given country,
whereas fact (6) relates movements of the slope of spreads to measures of US equity or
credit market risks for example. This fact appears to be new in the sovereign default
literature, and is also consistent with a “first-hitting-time” model, in which the default
barrier depends on international financial market conditions. I document it in the online
appendix and test whether my model generates this behavior of the term structure of
spreads.
(7) Holding-period expected excess returns on foreign currency sovereign debt increase with
the time-to-maturity of the credit instrument; in addition, most of the excess return
differential between short term bonds and longer term bonds is earned in “crisis” periods
– defined by Broner, Lorenzoni, and Schmukler (2013) as a period when the level of credit
spreads for the countries of interest are greater than the previous quarterly average plus
8
300bps. I will provide additional evidence supporting this result in the online appendix,
by focusing on CDS as opposed to bonds. I will also emphasize that this excess return is
actually earned during periods of high risk prices – which can be interpreted as periods
during which international debt investors are more risk-averse than usual. This fact is
consistent with a sovereign debt “risk exposure” that is increasing with the maturity of
the debt instrument.
(8) Holding-period excess returns on sovereign bonds are higher for countries with worse
credit ratings. Borri and Verdelhan (2011) document this fact by looking at returns on
sovereign bonds in the EMBI index, and I provide additional evidence by looking at CDS
returns. Whereas Borri and Verdelhan (2011) argue that they would need a new source
of exogenous country heterogeneity in order to account for fact (8), I will argue in the
paper that such fact arises because the “risk exposure” of sovereign credit instruments
is higher after a country has been hit by a sequence of bad fundamental shocks.
In the next sections I leverage the canonical framework of Eaton and Gersovitz (1981),
Arellano (2008) and Aguiar and Gopinath (2006) in order to build a continuous time model
of sovereign defaults where international capital markets take on a prominent role. I will
then confront the resulting model to the stylized facts discussed above.
3 A Continuous Time Sovereign Default Model
3.1 The Government
While I focus my empirical and quantitative analyses on the credit risk of different sovereign
governments, the theoretical section of this paper only deals with a single government “n”.
For simplicity, I abstract from interactions that different countries may have (such as cross-
border trade flows), except through a common marginal investor in their sovereign debt. I
thus abstract from the identity of the government in my notation. Country n is endowed with
real output Yt per unit of time, which evolves according to a Markov modulated geometric
Brownian motion:dYtYt
= µstdt+ σst · dBt (1)
My notation will use bold letters for vectors. Btt≥0 is an Nb-dimensional standard Brown-
ian motion on the underlying probability space (Ω,F ,P). I use multi-dimensional Brownian
shocks to be able to discuss how idiosyncratic, regional and global shocks affect spread and
return properties of sovereign debt. stt≥0, taking values in 1, ..., Ns, is a discrete state
Markov process with a generator matrix Λ = (Λij)1≤i,j≤Ns that is assumed to be conservative
9
(in other words∑Ns
j=1 Λij = 0 for all i). I will assume that stt≥0 is recurrent, thus admitting
a unique stationary distribution π (an Ns×1 real-valued positive vector) that solves π′Λ = 0,
and whose elements sum to 1. I will note N(i,j)t the Poisson counting process for transitions
from state i to state j. I will refer to P as the physical probability measure, and note Ft the
σ-algebra generated by the Brownian motion Bt and the discrete state Markov process st.
The Markov state st is not essential for modeling the country’s output dynamics – in
most of the quantitative applications of this model, I will in fact assume that the expected
GDP growth rate and the GDP growth volatility do not depend on the regime st. It could
also be argued that the length of the GDP growth time series of the countries of interest is
too limited to detect such regime shifts in the data7. Instead, the discrete regime st will be
the key variable describing the state of the creditors’ stochastic discount factor, as will be
discussed in Section 3.2. I keep the flexibility to model a country’s output dynamics as a
Markov modulated geometric Brownian motion for two reasons. First, it allows me to deal
with time-varying output growth volatility, a phenomenon empirically relevant for certain
countries, as Seoane (2013) suggests when focusing on Greece, Italy, Spain and Portugal.
Second, I argue in Section A.1.2 that this stochastic growth model enables me to approximate
the output process used by Aguiar and Gopinath (2006) and many other articles in the
international macroeconomic literature8. Lastly, I show in Section 8 that a mean-reverting
output growth rate can be obtained endogenously by introducing capital accumulation and a
simple “AK” production technology. The government objective is to maximize the life-time
utility function:
Jt = E[∫ +∞
t
ϕ (Cs, Js) ds|Ft]
(2)
The notation E denotes expectations under the measure P. The aggregator ϕ takes the
following form:
ϕ (C, J) := δ1− γ1− ρ
J
(C1−ρ
((1− γ)J)1−ρ1−γ− 1
)(3)
This preference specification is a generalization of the standard time-separable iso-elastic
preferences to a non-time-separable framework, where intertemporal substitution can be de-
7For most countries of interest, I have yearly GDP growth data since 1970 – in other words, approximately40 data-points. Estimating a Markov-switching output model with 2 Markov states for example would requireestimating 2 expected growth rates, 2 growth volatilites, and 2 transition probabilities, leading to pointestimates likely to have large standard errors.
8Other articles that use this output process include, amongst others, Borri and Verdelhan (2011), Aguiaret al. (2016b), Aguiar and Gopinath (2004). In those articles, log output has a unit root, and output growthis a stationary and mean-reverting process. Viewed differently, the martingale decomposition of log output(see Gordin (1969)) features (a) a time trend, (b) a martingale component with constant volatility and (c)a stationary component that is the sum of two Ornstein-Uhlenbeck processes, one of them fully correlatedwith the martingale component, while the other is independent.
10
coupled from risk-aversion. δ is the government rate of time preference, 1/ρ is the inter-
temporal elasticity of substitution, while γ is the risk aversion coefficient. The standard
iso-elastic time-separable preference specification corresponds to the parameter restriction
γ = ρ. In such case, the life-time utility of the government takes the more familiar form:
Jt = E[∫ +∞
t
δe−δ(s−t)C1−γs
1− γds|Ft
](4)
If the government does not have any financial contracts at its disposal, its life-time utility is
equal to:
Jst(Yt) = KstY1−γt (5)
Equation (5) as well as the Ns constants Kii≤Ns are determined in Section A.1.3. In order
for equation (5) to be well defined, I need to impose a parameter restriction that will be
assumed going forward.
Assumption 1. Let Aii≤Ns be the family of constants defined via:
Ai := δ + (ρ− 1)(µi −1
2γ|σi|2) (6)
Then (δ, ρ, γ, µii≤Ns , σii≤Ns) are such that Ai > 0 for all i.
The government does not have a full set of Arrow-Debreu securities at its disposal. In-
stead, it can only use non-contingent long-term debt contracts, with aggregate face value
Ft and coupon rate κ. The incentive for the government to issue debt is two-fold: first,
it enables the government to smooth consumption, and to reduce the welfare losses associ-
ated with consumption volatility. Second, differences between the government’s rate of time
preference and sovereign debt investors’ discount rates will enable the government to “tilt”
consumption into the present.
During each time period (t, t+dt], a constant fraction mdt of the government’s total debt
amortizes, which the government repays with mFtdt units of output. This contract structure
guarantees a constant debt average life of 1/m years, and allows me to carry only one state
variable (Ft) as a descriptor of the government’s indebtedness, as opposed to the full history
of past debt issuances. The long-term debt assumption is also essential in my continuous
time framework in order to insure that an equilibrium with default can be supported: I show
in Section A.1.1 that the continuous sample paths of my output process preclude short term
debt from being supportable in any sovereign default equilibrium. During each time period
(t, t + dt], the government can also decide to issue a dollar face amount Itdt of bonds. This
formulation of an admissible issuance policy prevents “lumpy” debt issuances, and results in
11
a government face value process Ft that is absolutely continuous:
dFt = (It −mFt) dt (7)
Per period flow consumption consists of (a) total per-period output, plus (b) proceeds (in
units of consumption goods) raised from capital markets minus (c) debt interest and principal
repayments due:
Ct = Yt + ItDt − (κ+m)Ft (8)
In the above, Dt is the endogenous debt price per unit of face value, and is determined in
equilibrium. My formulation of the debt dynamics as well as the resource constraint for the
government lead to a cumulative consumption process that is absolutely continuous; in other
words, the government does not consume in “lumpy fashion”, but rather always in “flow”
fashion. I can interpret the difference Yt − Ct as the trade balance. The government cannot
commit to repay its debt, which is thus credit risky. In other words, the government will
choose a sequence of default times τkk≥1 out of the set of sequences of stopping times9.
Default leads to the following consequences. First, output jumps down, from Yτ− to Yτ =
αYτ−, with α < 1. Second, the country is locked out of capital markets for a (random) time
period τe that is exponentially distributed with parameter λ. Once the country emerges from
financial autarky, it has an outstanding debt balance that is only a fraction of its pre-default
value, according to:
Fτ+τe = θYτ+τe
Yτ−Fτ− (9)
One can think of the parameter θ as the outcome of a bargaining game between creditors
and the sovereign government, once such government has elected to default. However, for
simplicity and since the strategic interactions between the government in default and its
creditors are not a focus of this paper, I elect to model the outcome of this renegotiation
exogenously10.
9The continuous time setting of this model allows me to abstract from the specific timing assumption ofthe government bond auction. In discrete time models, Cole and Kehoe (1996), Aguiar and Amador (2013)and Aguiar et al. (2016b) (for example) all assume that the bond auction happens before the default decisionis made by the government, while Aguiar and Gopinath (2006), Arellano (2008) and many other quantitativemodels of sovereign debt assume that the government makes its default decision before the bond auctiontakes place. The former timing convention allows, in discrete time, for the existence of potentially multipleequilibria, induced by the creditor’s self-fulfilling belief that the government will default immediately afterdebt has been issued, leading to a low auction debt price and a rational decision by the government to default.Those considerations are absent from the continuous time environment.
10Note that the adjustment factorYτ+τeYτ−
in the debt face value post-restructuring is included for tractability
purposes, since it will lead me to solve nested ordinary differential equations, as opposed to integro-differentialequations. This feature is used in Nuno and Thomas (2015).
12
3.2 Creditors
External creditors purchase the debt issued by the government. I model their marginal utility
process Mt (which I will also refer to as the stochastic discount factor, or “SDF”) as a random
walk with two independent components – a diffusion component, and a jump component.
More specifically, Mt evolves according to:
dMt
Mt−= −rstdt− νst · dBt +
∑st 6=st−
(eυ(st−,st) − 1
) (dN
(st−,st)t − Λst−,stdt
)(10)
Conditioned on being in the discrete Markov state i, creditors’ risk free rate ri and the Nb×1
risk price vector νi are constant. As Section A.1.4 or Chen (2010) show, this stochastic
discount factor can be obtained for example if creditors have iso-elastic time-separable or
recursive preferences and an equilibrium consumption process that follows a Markov mod-
ulated geometric Brownian motion. This stochastic discount factor can also be obtained in
a general equilibrium environment with a continuum of countries, by re-intrepreting Ct as
spending by government n, Yt as the tax revenues of government n, and introducing a “world
investor” who can diversify away all countries’ idiosyncratic risks, as I show in Section A.1.5.
This latter interpretation has the benefit of tying the world interest rate and the world risk
prices to the investor’s preferences and the countries’ endowment growth rates, but would
not add any additional insight to the paper. Finally, as explained in Section A.1.4, the jth
coordinate of νi represents the excess return compensation per unit of jth Brownian shock
earned by investors – hence why I refer to νi as the vector of risk prices in state i. Similarly,
(eυ(i,j)−1) is the jump-risk premium earned by investors per unit of jump risk, in connection
with shifts from SDF state i to SDF state j.
My formulation of the stochastic discount factor implicitly assumes that government
n’s sovereign debt component of the creditor’s portfolio is negligible, and that government
n’s sovereign debt cash-flows do not alter the equilibrium consumption of creditors. This
assumption seems reasonable: according to the World Bank, the aggregate external debt of
emerging market countries was approximately $1tn in 2014; while economically large, this
quantity is small compared to the $19tn market capitalization of stocks traded on the NYSE,
the $7tn market capitalization of stocks traded on the Nasdaq, and the $35tn size of the US
bond market.
Given my assumed investor pricing kernel, any Ft+s-measurable amount At+s received at
time t + s will be valued by investors by weighting such future cash-flow by the investors’
future marginal utility, and taking expectations. One can also use a standard tool of the
financial economics literature, and instead discount this future cashflow At+s at the risk-free
13
rate, while distorting the probability distribution of such future cashflow via the following
change in measure:
Pricet (At+s) = E[Mt+s
Mt
At+s|Ft]
:= E[e−
∫ s0 rt+uduAt+s|Ft
]E is the risk-neutral expectation operator. It implicitly defines the risk-neutral measure Q,
under which Bt := Bt+∫ t
0νsudu is a standard Nb dimensional Brownian motion, and under
which stt≥0 is a discrete state Markov process with generator matrix Λ, whose (i, j) element
is Λij = eυ(i,j)Λij, for i 6= j11.
Since most of the elements of the model have been introduced, I conclude this section by
introducing two parameter restrictions. The first restriction guarantees that the risk-neutral
value of a claim to the government’s output be finite.
Assumption 2. (rii≤Ns , νii≤Ns , µii≤Ns , σii≤Ns) jointly satisfy the following param-
eter restriction:
ri + νi · σi − µi > 0 ∀i ∈ 1, ..., Ns (11)
The second restriction insures that the government is impatient enough to front-load
consumption in equilibrium. To be specific, when the government has neither debt nor assets
outstanding, I need the government’s financing policy to be such that it wants to borrow,
instead of save. While I do not provide an explicit restriction on the deep model parameters
in order to satisfy such condition, I verify ex-post after solving the model that it is the case.
Intuitively, this parameter restriction should insure that the rate of time-preference δ of the
government is sufficiently greater than the level of interest rates at which the government
can finance itself via debt issuances.
3.3 Debt Valuation, Government Problem and Equilibrium
In this section, I focus on a Markovian setting and define admissible issuance and default
policies of the government. Any admissible issuance and default policy will give rise to
controlled Markov processes for the GDP and the debt face value. I then define the sovereign
debt price and the life-time utility of the government, discuss the stochastic control problem of
the government, and define a Markov perfect equilibrium. All technical details are relegated
to the appendix, in Section A.1.6.
11Λ is also assumed to be conservative.
14
The payoff-relevant variables for the sovereign government and creditors are st, Yt and
Ft. The state space will be 1, ..., Ns × R2, or a subset thereof. An admissible issuance
policy I will be a set of Ns functions Ii(Y, F ) that satisfy a particular integrability condition,
and an admissible default policy τ will be a sequence of increasing stopping times τkk≥1
that can be written as first hitting times of a particular subset of the state space. I will also
note τe,kk≥1 the sequence of capital markets’ re-entry delays, in other words the sequence
of independent exponentially distributed time lengths spent by the country in autarky. I will
note I the set of admissible issuance policies, and T the set of admissible default policies.
For any given admissible default policy τ ∈ T , there is an associated controlled output
process Y (τ ), which follows equation (1) at all times except when a default occurs, at which
point Y (τ ) suffers a downward jump. For any given admissible issuance policy I ∈ I, and
default policy τ ∈ T , there is an associated controlled debt face value process F (I,τ ), which
follows equation (7) at all times except when a default occurs, at which point the aggregate
debt face value stays unchanged, until reset at a lower level according to equation (9).
Creditors price the sovereign debt rationally. If they anticipate that the government will
follow admissible policy (I, τ ) ∈ I × T , they will value one unit of debt of a government
currently performing under its contractual obligations as follows:
Di (Y, F ; (I, τ )) := Ei,Y,F[∫ τ
0
e−∫ t0 (rsu+m)du(κ+m)dt
+e−∫ τ0 (rsu+m)duDd
sτ
(Y
(τ )τ− , F
(I,τ )τ− ; (I, τ )
)](12)
The stopping time τ in the equation above refers to the first element of the sequence of default
times τ . The superscript notation next to the expectation operator denotes the conditioning
on the initial state. Ddi (·, ·; (I, τ )) is the debt price in default, which satisfies:
Ddi (Y, F ; (I, τ )) := Ei,Y,F
[e−
∫ τe0 rsudu
F(I,τ )τe
FDsτe
(Y (τ )τe , F (I,τ )
τe ; (I, τ ))]
(13)
The stopping time τe in equation (13) refers to the first capital markets’ re-entry delay of the
sequence τe,kk≥1. I use a notation that makes the dependence of the debt price functions
on the anticipated issuance and default policies explicit. Equations (12) and (13) can be
interpreted as follows: creditors receive cash-flows κ+m per unit of time on a debt balance
that amortizes exponentially at rate m. Following a default, creditors receive no cash-flows
for the exponentially distributed random time τe, following which their claim face value suffers
a haircut. The expectations are taken under the risk-neutral measure Q.
I then focus on the government life-time utility. Given a debt price schedule D :=
15
Di(·, ·)i≤Ns that the government faces, and given admissible issuance and default policies
(I, τ ) used by the government (where (I, τ ) might not necessarily be consistent with the debt
pricesD), there is a controlled flow consumption process C(I,τ ;D)t , which satisfies equation (8)
when the government is performing, and which is equal to output whenever the government
is in default. This leads to the following government life-time utility:
Ji (Y, F ; (I, τ );D) = Ei,Y,F[∫ ∞
0
ϕ(C
(I,τ ;D)t , Jst
(Y
(τ )t , F
(I,τ )t ; (I, τ );D
))dt
](14)
In the time-separable preference case, the life-time utility takes the more familiar form:
Ji (Y, F ; (I, τ );D) = Ei,Y,F
∫ ∞0
δe−δt
(C
(I,τ ;D)t
)1−γ
1− γdt
(15)
In both cases, the expectations are taken under the physical probability measure P. The
government takes as given the family of debt price functions D and Dd and chooses its
issuance and default policies in order to solve the following problem:
Vi(Y, F ;D) := sup(I,τ )∈I×T
Ji (Y, F ; (I, τ );D) (16)
When choosing its issuance policy, the government takes into account the debt price schedule
and the impact that such schedule has on flow consumption, via the resource constraint.
Consistent with Maskin and Tirole (2001), I then define a Markov perfect equilibrium as
follows.
Definition 1. A Markov perfect equilibrium is a set of Markovian issuance and default
policies (I∗, τ ∗) ∈ I × T such that for any initial state (i, Y, F ),
(I∗, τ ∗) = arg max(I,τ )∈I×T
Ji (Y, F ; (I, τ );D (·, ·; (I∗, τ ∗)))
For a given equilibrium (I∗, τ ∗), I will note Vst(Yt, Ft) the government’s equilibrium value
function when performing, and V dsτ (Yτ−, Fτ−) the government’s equilibrium value function
at default time τ , when the pre-default output is equal to Yτ− and the pre-default debt face
value is equal to Fτ−. The following set of lemmas will help narrow down the class of Markov
perfect equilibria I will be focusing on.
16
Lemma 1. If for each state i ≤ Ns, the debt price schedule Di(·, ·) is homogeneous of degree
zero and decreasing in F , then the life-time utility Vi (·, ·;D) is strictly increasing in Y and
strictly decreasing in F . In such case, the optimal issuance policy is homogeneous of degree
one and the optimal government default policy is a state-dependent barrier policy, in other
words there exists a set of positive cutoffs xii≤Ns such that τk+1 = inft ≥ τk + τe,k : Ft ≥xstYt (with τ0 = τe,0 = 0). Finally, the life-time utilities Vi (·, ·;D) are homogeneous of
degree 1− γ.
The proof of this lemma is detailed in Section A.1.7. I then focus on the debt price
schedule for specific types of issuance and default policies.
Lemma 2. If I ∈ I is a homogeneous of degree 1 Markov issuance policy, and if τ ∈ T is a
barrier default policy, the debt price functions Di (·, ·) are homogeneous of degree zero and
decreasing in F .
The proof can be found in Section A.1.8. As discussed in the next section, by restricting
the set of equilibria of focus, Lemma 1 and Lemma 2 will enable me to reduce the dimension-
ality of the state space and deal with only one continuous and one discrete state variables.
3.4 Equilibrium Debt Value
Using the previous observations, I look for an equilibrium of the model for which xt := Ft/Yt
(the debt-to-output ratio) and st are the unique state variables, and for which the government
follows a barrier policy: it defaults when the debt-to-output ratio xt is at or above a state-
dependent threshold xst . In other words, the sovereign’s first time of default is τ := inft ≥0 : xt ≥ xst. The government issuance policy can be re-written It = ιst(xt)Yt, where
ιst(xt) represents the rate of debt issuance per unit of output, for a given debt-to-output
ratio and when the discrete Markov state is st. ι > 0 means that the government is either
decumulating net foreign assets (when x < 0) or borrowing (when x > 0), whereas ι < 0
means that the government is buying back outstanding debt. The dynamic evolution of the
controlled stochastic process xt (under the measure P) when the government is performing
under its debt obligations is as follows:
dx(ι,τ )t =
(ιst
(x
(ι,τ )t
)−(m+ µst − |σst|2
)x
(ι,τ )t
)dt− x(ι,τ )
t σst · dBt
The debt-to-GDP ratio increases with the issuance rate ιt and with the Ito term |σst|2xt,and decreases thanks to GDP growth µstxt and debt amortizations mxt. Under the risk-
neutral measure Q, following Girsanov’s theorem, the drift of xt must be adjusted upward
17
by νst · σstxt. Creditors take the government issuance policy ι and the government default
policy as given when pricing a unit of sovereign debt. Finally, I will postulate (and verify)
that in equilibrium, ιi(0) > 0 for all states i ≤ Ns. This means that when the government has
neither financial assets nor financial liabilities, it finds it optimal to borrow and front-load
consumption in all states i ≤ Ns. This also means that once the state xt enters the interval
(0,maxi xi), it never leaves such interval, since the diffusion term in the stochastic differential
equation for xt vanishes and the drift term is strictly positive. I thus restrict the focus of my
analysis to the state space 1, ..., Ns × (0,maxi xi).
An illustration of the state space, as well as a realization of the output, SDF state and
the debt face value paths, is illustrated in Figure 1a, with the corresponding evolution of the
state variables xt and st in Figure 1b. Defaults occur at times τ1 and τ2. The length of time
spent in autarky after the ith default is τe,i, after which the aggregate debt face amount is
reset at a fraction of its pre-default value.
With an abuse of notation, I use Di (x; (ι, τ )) (resp. Ddi (x; (ι, τ ))) to denote the debt
value (resp. the debt value in default) per dollar of face-value when the debt-to-output ratio
is x and the SDF regime is i. I will also omit the dependence of the debt price function on
the government policies (ι, τ ) whenever possible. When st is in state i and when x ∈ [0, xi),
the government is indebted, and the price Di (·; (ι, τ )) of defaultable sovereign debt verifies:
Di(x) = Ei,x[∫ τ
0
e−∫ t0 (rsu+m)du(κ+m)dt+ e−
∫ τ0 (rsu+m)duDd
sτ (xτ−)
](17)
Using Feynman-Kac, it is immediate to show that Di is twice differentiable and satisfies the
following HJB equation for x ∈ (0, xi):
(ri +m)Di(x) = κ+m+ L(ι)i Di(x) +
Ns∑j=1
ΛijDj(x) (18)
For ease of notation, I have introduced the infinitesimal operator L(ι)i as follows:
L(ι)i :=
[ιi(x)−
(µi +m− |σi|2 − νi · σi
)x] ∂∂x
+1
2|σi|2x2 ∂
2
∂x2
2×Ns boundary conditions are required in order to solve this set of Ns nested second order
18
t
outp
utY
and
deb
tF
τ1 τ2
SD
Fst
ates
output Ytdebt FtSDF state st
(a) Output Yt and Debt Ft
x1
x2
x3
τ1
τe,1
τ2
τe,2
t
x
debt/GDP xtdefault boundary xst
(b) Debt-to-GDP ratio xt
19
ordinary differential equations. They are as follows, for 1 ≤ i ≤ Ns:
Di(xi) = Ddi (xi) (19)
(ri +m)Di(0) = κ+m+ ιi(0)D′i(0) +Ns∑j=1
ΛijDj(0) (20)
For each state i, the first boundary condition is a value matching condition, which says that
the debt price at the default boundary x = xi is equal to the price of a claim on the defaulted
debt, Ddi (xi) (which will be calculated later on). The second boundary condition is a Robin
boundary condition; it relates the value of the function Di at the origin to its first derivative
at the origin. It can be obtained by simply taking a limit of the HJB equation satisfied by
Di at x = 0. I need to compute the debt price in default Ddi (x), for x ≥ xi and 1 ≤ i ≤ Ns.
Assume that at time of default τ , the state is sτ = i. When the country exits financial
autarky, its debt-to-GDP ratio is equal to Fτ+τe
Yτ+τe= θFτ−
Yτ−= θxτ−. Note that it is possible
that xτ− > xsτ when the sovereign defaults. This happens upon the occurrence of a “jump-
to-default”, in other words a situation where the discrete SDF state jumps from sτ− = j to
sτ = i and when xi < xτ− < xj. Thus, I have the following for x ≥ xi:
Ddi (x) = Ei
[exp
(−∫ τe
0
rst+udu
)Ft+τeFt−
Dst+τe (θx)
]Section A.1.9 establishes the following formula for the defaulted debt price:
Dd(x) = λθαΞ−1D(θx) (21)
In equation (21), Dd(x) is the Ns× 1 vector with ith element Ddi (x), and the Ns×Ns matrix
Ξ := diagi (ri + νi · σi + λ− µi) − Λ is well defined thanks to Assumption 2. Finally, note
that this equation is valid for each coordinate i for x ≥ xi.
I end this section by discussing two different aspects of the model. First, the existence of a
discrete number of SDF regimes leads to two types of defaults: defaults following a sequence
of bad GDP shocks, as well as defaults induced by jumps in the SDF state, from a state of
low risk prices to a state of higher risk prices. Both types are illustrated in Figure 1a and
Figure 1b. In this example, a default occurs at τ1, after a sequence of bad GDP shocks that
cause the debt-to-GDP ratio to breach the optimal default boundary that the government
has set in such SDF regime. In the same figure, a default occurs at τ2, triggered by a jump
in the SDF state. At such time, an SDF regime shift occurs, from sτ2− = 3 to sτ2 = 2,
and the debt-to-GDP ratio satisfies x3 > xτ2− > x2. In other words, before the SDF jump,
the debt-to-GDP ratio of the sovereign is below the optimal default boundary, but as the
20
SDF regime shifts, the debt-to-GDP ratio is suddenly greater than the new optimal default
boundary, causing the sovereign to immediately default. Since the SDF I use will price the
sovereign debt of multiple countries, SDF regime shifts induce correlated defaults amongst
sovereign governments. Note that jump-to-default risk exists even if the GDP growth rate
and GDP growth volatility are not regime-dependent – so long as the SDF exhibits different
risk prices in different regimes.
Second, note that when x 0, the government debt balance is negligible compared to
output. However, the price of any infinitesimally small unit of debt is actually not equal to the
risk-free debt price, since the debt price needs to factor in the dilution risk of the government,
whose optimal issuance policy will dictate to issue debt to front-load consumption. This
observation is in stark contrast with what happens in structural corporate credit risk models
(see for example Leland (1994) or Leland (1998)) – in those models, the firm can commit to
a financing policy (typically, maintaining the debt face value constant), but cannot commit
to a default policy, leading to a debt price that is equal to the risk-free debt price when the
level of fundamentals becomes arbitrarily large compared to the debt face value.
3.5 Equilibrium Debt Issuance and Default Policies
Now consider the government’s problem, as described in Section 3.3. As a reminder, the gov-
ernment takes the debt price schedule D(·) as given when solving its optimization problem.
Thanks to Lemma 1, the government value function in state i can be written as follows:
Vi(Y, F ) := vi(x)Y 1−γ (22)
In the above, the function vi will be positive when γ ∈ (0, 1), and negative when γ > 1.
Since Vi is decreasing in F , I also have the sign restriction v′i(x) < 0. An appropriate change-
in-measure described in Section A.1.10 shows that the HJB equation associated with the
government problem, in the continuation region [0, xi), is the following:
1− γ1− ρ
Aivi(x)−Ns∑j=1
Λijvj(x) =
supιi
[δ
(1 + ιiDi(x)− (κ+m)x)1−ρ [(1− γ)vi(x)]ρ−γ1−γ
1− ρ+ L(ι)
i vi(x)
](23)
In the above, I have used the differential operator L(ι)i defined as follows:
L(ι)i :=
[ιi −
(µi +m− γ|σi|2
)x] ∂∂x
+1
2|σi|2x2 ∂
2
∂x2
21
The optimal state-contingent issuance policy ιi is then given by:
maxιi
[δ
1− ρ(1 + ιiDi(x)− (κ+m)x)1−ρ [(1− γ)vi(x)]
ρ−γ1−γ + ιiv
′i(x)
]This yields the (necessary and sufficient, given the strict concavity of the expression in
brackets w.r.t. ιi) first order condition:
Di(x)δci(x)−ρ [(1− γ)vi(x)]ρ−γ1−γ = −v′i(x) (24)
In the above, I have introduced the consumption-to-GDP ratio ci := C/Y when the discrete
Markov state is i. Focusing on equation (24), I notice that the left-hand side is the product
of (a) the marginal utility of consumption δci(x)−ρ [(1− γ)vi(x)]ρ−γ1−γ and (b) the debt price,
while the right-hand side is the marginal cost of taking on one extra unit of debt. The optimal
Markov issuance policy function ιi(x) is given by:
ιi(x) =1
Di(x)
(δDi(x) [(1− γ)vi(x)]ρ−γ1−γ
−v′i(x)
)1/ρ
+ (κ+m)x− 1
(25)
The expression is well defined since I showed previously that v′i(x) < 0. The dependence
of the issuance policy on the model parameters or on the debt price schedule (which the
government takes as given) are ambiguous, since those issuance parameters will also have a
feedback effect on the felicity function and its derivative. I can however perform a “partial
equilibrium” analysis of the debt price schedule in the unit elasticity of substitution case, i.e.
when ρ = 1. In such case, ιi(x) is an increasing function of Di(x) whenever the sovereign
output Yt is greater than the total debt service owed (κ + m)Ft, which will always be the
case in equilibrium (in other words in equilibrium, the sovereign will have defaulted before
the sovereign output falls low enough that new debt issuances are required to service the
existing debt). For the case where the elasticity of substitution is different from 1, I verify
numerically that this comparative static result still holds: when the debt price schedule is
more beneficial to the sovereign, the latter takes advantage of it through additional issuances.
For a given set of default thresholds xii≤Ns , 2×Ns additional boundary conditions are
needed do solve the system of Ns equations (23). The first set of conditions relates to value
matching at the default boundary xi. Let V di (Y, F ) be the government value function in
default, if the pre-default output level is Y and the pre-default debt face value if F . I show in
Section A.1.11 that V di (Y, F ) = vdi (xi) (αY )1−γ, which leads to the following value-matching
condition:
vi (xi) = α1−γvdi (xi) (26)
22
vdi (x) solves a system of non-linear equations discussed in Section A.1.11. I also have a set of
Ns Robin boundary conditions, linking the value function at the origin to its derivative, via:
1− γ1− ρ
Aivi(0)−Ns∑j=1
Λijvj(0) = δ(1 + ιi(0)Di(0))1−ρ [(1− γ)vi(0)]
ρ−γ1−γ
1− ρ+ ιi(0)v′i(0) (27)
I finally focus on the optimal default policy. Since it is always an option for the government
to default, I must have Vi(Y, F ) − V di (Y, F ) ≥ 0 for all states (Y, F ). This leads to a set of
Ns smooth-pasting conditions:
v′i (xi) = α1−γ(vdi )′(xi) (28)
Section A.1.11 establishes more formally this optimality condition and shows how (vdi )′(xi)
can be expressed as a function of vdi (xi) and vi(θxi). I conclude this section by two propo-
sitions. First, I establish a standard verification theorem for the government value function.
I then discuss the existence of a Markov perfect equilibrium, subject to the existence of a
solution to a set of ordinary differential equations.
Proposition 1. For any family of decreasing functions Di : R+ →[0, Drf
i
], assume that
there exists a family of functions vi (·;D) ∈ C1 (R+) ∩ C2 (R+ \ xi), which satisfies for
1 ≤ i ≤ Ns:
0 = max
[supι
[−1− γ
1− ρAivi (x;D) +
Ns∑j=1
Λijvj(x;D)
+δ(1 + ιiDi(x)− (κ+m)x)1−ρ [(1− γ)vi (x;D)]
ρ−γ1−γ
1− ρ+ L(ι)vi (x;D)
];α1−γvdi (x;D)− vi (x;D)
],
where vd (x;D) satisfies (using the Ns ×Ns matrix Υ := 1−γ1−ρdiagi (Ai) + λI − Λ):
Υvd(x;D)− λv(θx;D) =δ
1− ρ[(1− γ)vd(x;D)
] ρ−γ1−γ ,
Then for any state i ≤ Ns and any x ∈ R+, vi(x;D) ≥ Ji(1, x; (ι, τ );D) for any (ι, τ ) ∈I × T that satisfy limt→+∞ inf e−
∫ t0
1−γ1−ρAsuduvst
(x
(ι,τ )t ;D
)≤ 0. Let the family of thresholds
xi1≤i≤Ns ∈ (R+)Ns satisfy:
(vd)′
(xi) = λθ
(Υ + δ
γ − ρ1− ρ
diagj
([(1− γ)vdj (xi)
]− 1−ρ1−γ))−1
v′(θxi)
23
Let (ι∗, τ ∗) be defined as follows:
ι∗i (x;D) :=1
Di(x)
(δDi(x) [(1− γ)vi (x;D)]ρ−γ1−γ
−v′i (x;D)
)1/ρ
+ (κ+m)x− 1
τ ∗(D) := inft ≥ 0 : xt ≥ xst
Then vi(x;D) = Ji(1, x; (ι∗, τ ∗);D) is the value function.
This proposition, proven in Section A.1.12, provides for a characterization of the optimal
issuance and default policies given a decreasing debt price schedule D. It does not establish
the existence of an equilibrium, which is achieved in the next proposition.
Proposition 2. Assume that there exists a set of functions vi(·)i≤Ns , Di(·)i≤Ns , and
a set of positive thresholds xii≤Ns such that the system of nested ordinary differential
equations (18), (23) subject to value-matching boundary conditions (19), (20), (26) and (27)
are satisfied, where ιi(·) satisfies equation (25) and each threshold xi satisfies the smooth
pasting condition (28). Then a Markov perfect equilibrium exists.
Proving the existence of a Markov perfect equilibrium without relying on the (strong)
assumptions of proposition 2 is beyond the scope of this paper, and I leave this proof for
future research. I provide in Section A.1.13 a discussion of the potential route to pursue to
establish such result. I also show in Section 4, for the particular case where ρ = γ = 0, that
a Markov perfect equilibrium exists, and it is unique in the class of “smooth” equilibria (i.e.
equilibria in which the debt face value process is restricted to being absolutely continuous).
3.6 Asset Pricing Moments
In this section, I discuss the implications of my model for the long term sovereign bond
spread, as well as excess returns earned by international investors on such bond. I also show
how to compute CDS premia and the excess return on these contracts.
3.6.1 Long Term Sovereign Debt Spreads
The sovereign bond spread ςi(x) is the constant margin over the risk-free benchmark that
is needed to discount the long-term sovereign bond’s cash flow stream assuming away any
default risk. In other words, the credit spread must verify:
Di(x) := Ei,x[∫ ∞
0
e−∫ t0 (rsu+ςi(x)+m)du(κ+m)dt
](29)
24
The credit spread ςi(x) is the unique positive solution to the following equation:
Di(x) = (m+ κ)
[(diagj (rj + ςi(x) +m)− Λ
)−1
1
]i
Using Ito’s lemma, credit spread innovations under P take the following form:
dςt − E [dςt|Ft] = −ς ′st−(xt)xtσst− · dBt +∑s′
(ςs′(xt)− ςst−(xt)
) (dN (st−s′) − Λst−s′dt
)(30)
What happens upon the occurrence of a GDP shock? Section A.1.15 establishes that ς ′i < 0
in any state i. Thus, good GDP shocks translate into decreases in sovereign bond spreads.
In other words, credit spreads are counter-cyclical in this model – a sequence of good GDP
shocks will on average lead to lower spreads, consistent with empirical fact (4).
I can then leverage equation (30) to compute the instantaneous sovereign bond spread
volatility:
σςt =
√x2t |σst−|2ς ′st−(xt)2 +
∑s′
Λst−s′(ςs′(xt)− ςst−(xt)
)2(31)
In a model without SDF regime shifts, sovereign spread volatilities are purely driven by
the macroeconomic fundamentals of a country (in the context of this model, the debt-to-
GDP ratio x). Instead, SDF regime shifts in my model induce an additional component to
sovereign spread volatilities. A separate testable implication emerges from equation (31):
spread volatilies tend to be higher when the sovereign government is close to its endogenous
default boundary. Indeed, I show in the appendix that under mild conditions, the function
xς ′i(x) is increasing, meaning that the component of sovereign spread volatility stemming
from Brownian shocks increases as the sovereign government approaches its default cutoff.
Both predictions will be tested as part of my model validation.
Equation (30) also illustrates the crucial importance of the different SDF regimes for cross-
sectional spread correlations: absent those regime shifts, pairwise local spread correlation
between two different sovereign governments would only stem from output correlation, which
is at odds with fact (2). If I index by “a” and “b” two countries, the instantaneous spread
correlation between those countries takes the following form:
corrt (ςa,t, ςb,t) =ς ′a,st−ς
′b,st−
xa,txb,tσa,st− · σb,st− +∑
s′ Λst−s′(ςa,s′ − ςa,st−
) (ςb,s′ − ςb,st−
)σςa,tσ
ςb,t
In the formula above, for all states i ≤ Ns, the function ς ′a,i is evaluated at xa,t and the
function ς ′b,i is evaluated at xb,t. When the SDF state jumps from a low risk price level s to
a high risk price level s′, if both countries’ output processes are positively correlated with
25
the risk price vector in all discrete Markov states, spreads for both country “a” and country
“b” jump up, meaning that(ςa,s′ − ςa,st−
) (ςb,s′ − ςb,st−
)> 0. The same reasoning holds upon
a jump from a high risk price state to a low risk price state. Thus, the second term in my
formula for spread correlations above is positive: spread correlations are induced by SDF
regime shifts. This gives my model the potential for being consistent with fact (2) – but
only to the extend my countries of interest have output processes whose correlation with the
vector of risk prices have the same sign.
3.6.2 Long Term Sovereign Debt Returns
I then compute sovereign debt excess returns. Debt excess returns over the time period
(t, t + dt] include capital gains dDt, coupon payments κdt and principal repayments mdt,
while the opportunity cost is rstdt and reinvestment costs are equal mdt. Thus, excess
returns (under the physical measure P) are equal to:
dRet : =
dDt + (κ+m)dt
Dt
− (rst +m)dt
Using Ito’s lemma and the HJB equation satisfied by the family of debt values Di(·)i≤Ns ,I obtain expected excess returns (per unit of time) and return volatilities that are equal to:
E [dRet |Ft] = −
[xtD
′st(xt)
Dst(xt)νst · σst +
∑s′
Λsts′
(Ds′(xt)
Dst(xt)− 1
)(eυ(st,s′) − 1
)]dt (32)
var [dRet |Ft] =
x2tD′st(xt)
2
Dst(xt)2|σst|2dt+
∑s′
Λsts′
(Ds′(xt)
Dst(xt)− 1
)2
dt (33)
Thus, sovereign bond investors are compensated for taking Brownian risk (the first term on
the right hand-side of equation (32)), as well as for taking regime jump risk (the second
term on the right hand-side of equation (32)). The expected excess return can be read as
(minus) the local covariance between (a) sovereign debt returns and (b) the creditors’ pricing
kernel. This risk compensation is similar to a standard two-factor asset pricing compensation.
Indeed, I can interpret−xtD′st (xt)Dst (xt)
as the market beta of sovereign debt w.r.t. the shock Bt,
while νst · σst is the sovereign output claim’s risk premium earned in connection with such
shock. Similarly, the jump compensation (the second term in equation (32)) can be re-written:
∑s′
Λsts′
(eυ(st,s′) − 1
)(Ps′Pst− 1
)︸ ︷︷ ︸
output claim’s premium for jump risk
Ds′ (xt)Dst (xt)
− 1
Ps′Pst− 1
︸ ︷︷ ︸
market beta of sovereign debt w.r.t. jump risk
26
In the above, Pi is the price of a claim to the output of country i. Using the vector notation,
P =[diagi (ri + νi · σi − µi)− Λ
]−1
1. Alternatively, one can interpret those formulas using
the terminology of Hansen (2012a) or Hansen (2012b): in such case, the expected excess
return in equation (32) is the sum-product of (time-varying) risk prices (νst for the Brownian
shocks and(eυ(s,s′) − 1
)for jump risks) and (time-varying) risk exposures (
−xtD′st (xt)Dst (xt)
σst for
Brownian shocks and(Ds′Dst− 1)
for jumps).
Equation (32) highlights the crucial role of the local covariance between risk prices and the
GDP process for the determination of expected excess returns. When I tie the investor’s SDF
to US consumption growth and US consumption volatility (as in Section A.1.4), risk prices
are equal to the product of (i) US investors’ risk-aversion times (ii) US consumption growth
volatility. But US output and consumption growth exhibit only mild levels of correlation
with emerging market economies’ output growth, as documented in table “Country-Specific
Macro Moments” in the online appendix. One might then ask how this model might explain
the high level of expected excess returns earned on emerging market sovereign risks. Even if
the risk price vector νst is not (locally) correlated with the country’s output process, expected
excess returns can be positive when risk prices are time-varying and co-move with sovereign
debt prices. For this latter effect to “bite”, the pricing kernel must feature jumps (i.e. some
of the υ(i, j)1≤i,j≤Ns must be non-zero); the introduction of different SDF regimes only does
not suffice in order to produce large model-implied expected excess returns when νst ·σst ≤ 0
in all states.
Note also that the risk exposure to Brownian shocks (and the corresponding sovereign
debt market beta) depends on the elasticity−xD′i(x)
Di(x)of the bond price function. It turns out
that in all my numerical computations, the debt price function Di(·) is a concave function,
which means that the sovereign debt’s risk exposure to Brownian shocks is increasing in
x. This leads to another implication of the model: sovereign expected excess returns are
increasing in the debt-to-GDP ratio, consistent with fact (8). This implication was also
indirectly tested by Borri and Verdelhan (2011) when sorting sovereign debt portfolios by (a)
rating and (b) “market betas”, if one interprets the rating as a noisy measure of the debt-to-
GDP ratio. But while Borri and Verdelhan (2011) argue in the model section of their paper
that they would need to introduce two sources of heterogeneity in order to recreate their
empirical observation, I argue that this is not necessary: not only different countries may
have different business cycle correlations with foreign investors’ risk prices, but also countries
may have different risk exposures.
To conclude this section on long term debt returns, the properties of sovereign bond
return volatilities and cross-country correlations should be identical to those of sovereign
spread volatilies and cross-country sovereign spread correlations since realized bond returns
27
between t and t + dt are (approximately) proportional to spread changes during that time
period.
3.6.3 Credit Default Swap Premia and Returns
To conclude this section, I define ςi(x, T ), the credit default swap premium for a T maturity
contract. Conceptually, such premium should, at the time the trade is executed, compensate
the writer of protection for expected losses to be suffered on the contract. Mathematically,
ςi(x, T ) is defined as follows:
ςi(x, T ) :=Ex,i
[1τ<Te
−∫ τ0 rsudu max
(0, 1−Dd
sτ (xτ−))]
Ex,i[∫ T∧τ
0e−
∫ t0 rsududt
] =Li(x, T )
Pi(x, T )
Li(x, T ) is the risk-neutral expected credit loss, while Pi(x, T ) is the risk-neutral present-value
of CDS premia. Both expected losses and expected CDS premia can be calculated using the
Feynman-Kac formula, by solving a set of partial differential equations with boundary con-
ditions discussed in Section A.1.16. Section A.1.16 also provides formula for computing
expected excess returns and conditional return volatilities of CDS contracts of different ma-
turities. I can then test whether the model-implied term structure of spreads is consistent
with facts (5) and (6), and whether the term structure of expected excess returns is consistent
with fact (8).
My model with multiple SDF regimes (inducing multiple default boundaries, one per
regime) is particularly convenient in analyzing short term CDS premia, and confronting
them with the data. Indeed, when the CDS contract maturity is arbitrarily small (i.e. when
T → 0), default risk only stems from the risk of regime shifts. Under the assumption that
the discrete SDF states are ordered (i.e. under the assumption that x1 ≤ ... ≤ xNs), I then
have the following lemma, characterizing short term CDS premia.
Lemma 3. When the contract maturity T becomes arbitrarily small, the sovereign CDS
premium converges to the following limit:
limT→0
ςi(x, T ) =
0 if x ≤ minj xj or i = 1∑i−1j=1 Λij
(1−Dd
j (x))
otherwise
Lemma 3 shows that premium compensation for writers of short term sovereign CDS only
comes from SDF jump risk, as opposed to output volatility risk. In other words, a model that
28
does not feature multiple discrete SDF regimes would not feature high short-term market-
implied hazard rates, and would thus be inconsistent with stylized fact (3).
3.7 Macro Moments
One distinguishing feature of emerging market economies is the fact that the ratio of (a)
consumption growth volatility over (b) output growth volatility is substantially greater than
one (see for example Neumeyer and Perri (2005) or Aguiar and Gopinath (2004)). In my
model, as established in Section A.1.17, this ratio takes the following expression:
stdev[dCtCt
∣∣Ft]stdev
[dYtYt
∣∣Ft] =
√√√√(1−xtc′st(xt)
cst(xt)
)2
+1
|σst|2∑s′
Λst,s′
(cs′(xt)
cst(xt)− 1
)2
(34)
This ratio will thus crucially depend on how the consumption-to-output ratio ci(·) varies with
the debt-to-GDP ratio. As will be seen, as the debt-to-GDP ratio nears the default boundary,
the government will adjust its issuance policy downwards, meaning that the consumption-
to-output ratio ci(·) will be a decreasing function of x. This leads to a consumption growth
volatility that will be greater than output growth volatility. Note also that such volatility is
exacerbated by SDF regime shifts.
4 A Useful Benchmark: Risk-Neutral Goverment
The model presented above will need to be solved numerically, but there is one particular
parameter configuration where closed-form solutions are available: the case where the gov-
ernment is risk-neutral. Leveraging an insight from DeMarzo and He (2014), I develop in
Section A.1.18 the solution for this problem in the case where ρ = γ = 0. This benchmark is
useful to provide some intuition on the mechanics of the model.
Proposition 3. Assume that the government is risk-neutral, i.e. that γ = ρ = 0. Assume
that the risk-free rate in all SDF states is strictly less than the government’s rate of time
preference, i.e. ri < δ for all i ≤ Ns. Assume also that the GDP process is not regime-
specific, in other words assume that for all state i ≤ Ns, µi = µ and σi = σ. In such case,
there exists an equilibrium where the life-time government value function Vi(Y, F ), the debt
price schedule Di(Y, F ) and the optimal default cutoff xi are independent of the SDF state,
29
and have the following expressions:
D(x) =
(κ+m
δ +m
)[1−
(1− αθλ
δ+λ−µ
1− αθξλδ+λ−µ
)(xx
)ξ−1]
(35)
v(x) = δ
[1
δ − µ
(1−
(1− α
1− αθξλδ+λ−µ
)(xx
)ξ)− xD(x)
](36)
x =ξ
ξ − 1
(δ +m
κ+m
)( 1−αδ−µ
1− αθλδ+λ−µ
)(37)
In the above, ξ > 1 is a constant that only depends on the model parameters δ, µ,σ,m, and
not on the level of interest rates or the prices of risk. The scaled optimal financing policy
ιi(x) is SDF state-dependent and has the following expression:
ιi(x) =δ − riξ − 1
[(1− αθξλ
δ+λ−µ
1− αθλδ+λ−µ
)( xx
)ξ−1
− 1
]x− νi · σx (38)
The financing policy ιi(·) is a strictly decreasing function of x if δ+m > |σ|2− (m+µ), and
is otherwise hump-shaped. Conditional expected excess returns on the long-term bond have
the following expression:
E [dRet |Ft] =
ξ − 1(1− αθξλ
δ+λ−µ1− αθλ
δ+λ−µ
)(xx
)ξ−1 − 1
νi · σ (39)
The equilibrium above is the unique Markov perfect equilibrium featuring an absolutely
continuous debt face value process.
The first – and seemingly surprising – result of Proposition 3 is that the (output-normalized)
welfare value of a government without any debt outstanding (i.e. v(0)) is exactly equal to
the autarky welfare δ/(δ− µ). In other words, the fact that the risk-neutral government has
the option to take on debt financed by creditors with a discount rate that is strictly less than
the rate of time preference of the government is not welfare-improving for such government.
Similarly, when the government is indebted, the welfare of the government can be expressed
as the sum of (a) the welfare of a debt-free government that suffers a downward GDP drop
of (1 − α) % each time the state variable xt hits the boundary x, minus (b) the aggregate
value of sovereign debt, computed as if creditors were risk-neutral with a discount rate δ.
Importantly, neither the risk-free rates rii≤Ns , nor the price of risk νii≤Ns influence the
30
welfare value v(x):
V (Y, F ) = δ
Y
δ − µ
(1−
(1− α
1− αθξλδ+λ−µ
)(xx
)ξ)︸ ︷︷ ︸
value of “credit-risky” endowment
− FD(Y, F )︸ ︷︷ ︸aggregate debt value
This result is tightly related to the conjecture made in Coase (1972), and formally proven by
Stokey (1981) and Gul, Sonnenschein, and Wilson (1986), who show that a monopolist with
constant marginal costs selling a durable good to a continuum of consumers will actually
behave competitively, in the continuous-time limit, and not extract any monopoly rent. The
argument, in the context of the sovereign default model with a risk-neutral government,
works as follows: without commitment, no matter how many bonds the government sold in
the past, the government will sell more bonds if there are marginal gains from doing so (in
other words if those bonds can be sold at an implied interest rate strictly less than δ, i.e.
if δD(Y, F ) > −∂FV (Y, F )). But investors perfectly anticipate this behavior and thus price
the bonds at an implied interest rate of δ, therefore stripping away any potential welfare gain
that the government may extract from facing financiers that discount cash flows at a rate
strictly less than δ.
Note that this result is purely due to the continuous-time nature of my model; as high-
lighted by Stokey (1981) and as (unreported) computations illustrate, the discrete time coun-
terpart to this model would yield strictly positive welfare gains for the risk-neutral govern-
ment. Having a non-zero time period during which the government can commit not to issue
bonds is crucial in obtaining such result. Similarly, in the case of a risk-averse government,
the result above will no longer hold: the concavity in the flow payoff function will be such
that the government will extract welfare gains from issuing bonds to investors whose implied
interest rate is lower than the government’s rate of time preference. The country’s welfare,
sovereign bond prices and default boundaries will again depend on the level of risk-free rates
and prices of risk: since the government dislikes high levels of consumption volatility, its
financing policy will not fully adjust to keep sovereign bond prices unchanged; instead, the
adjustment will be partial. This result is analogous to what is showed theoretically in Kahn
(1986) in the context of the durable goods monopoly problem: rents can be extracted by the
monopolist if its marginal production costs are increasing.
As a consequence of Proposition 3, changes in the supply-side of capital lead the gov-
ernment to adjust its financing policy in such a way that sovereign bond prices and welfare
remain unchanged: with higher risk-free rates or higher prices of risk12, the higher financing
12This discussion assumes that the price of risk is positively correlated with the GDP process in all SDF
31
costs borne by the government are exactly compensated by a lower pace of debt issuances.
This translates into an upward adjustment of the country’s current account and trade bal-
ances, causing an endogenous sudden stop. In the context of an increase in the price of risk,
the magnitude of the current account reversal is high when the debt-to-GDP ratio of the
country is high. The financing policy of the government is also interesting to study since it
highlights the fact that when bond investors are risk-neutral (i.e. when the price of risk is
identically zero), the issuance policy of the government is always positive: it is never efficient
for the government to buy back debt. This result echos an insight from Bulow, Rogoff, and
Dornbusch (1988) who show, in the context of a one-period model of sovereign default with
a risk-neutral government and risk-neutral lenders, that it is never welfare-improving for a
country to buy back its own debt. This result breaks down in the presence of risk-averse
lenders, whose price of risk has a positive correlation with the country’s endowment process:
in such case, Equation (38) shows that there are parameter configurations where, for high
debt-to-GDP ratios, the country does find it optimal to buy back its own debt. For this to
be the case, the parameters of the model need to satisfy the following condition:
αλθ(1− θξ−1
)δ + λ− µ− αθλ
<ξ − 1
δ − riσ · νi
In other words, if either the price of risk |νi| is sufficiently high, or the GDP drop upon
default 1−α is sufficiently severe, or the risk-free rate ri is sufficiently close to the government
discount rate δ, it is sometimes optimal for the government to buy back debt. The key to this
result is the fact that the probability measure under which investors discount cash-flows (the
“risk-neutral” measure) is different from the probability measure (the “physical” measure)
under which the government optimizes. A different interpretation of this result can be put
as follows: persistent differences in beliefs about the growth rate of the country’s endowment
(where investors would be more “pessimistic” than the government) would also lead the
government to buy back debt when the debt-to-GDP ratio of the country is high.
Finally, the expected excess return earned on sovereign bonds is increasing in the debt-to-
GDP ratio, confirming the ability of the model to replicate – at least qualitatively – empirical
fact (8). In environments with constant risk-free rates, while an increase in the price of risk
induces a sudden stop, sovereign bond prices remain unchanged, leading to sovereign credit
spreads that are not reacting to these worse capital market conditions. Finally, the closed-
form expressions of Proposition 3 allow me to derive the following comparative static results.
Corollary 1. The default boundary x is decreasing in the impatience parameter δ, de-
states (i.e. νi · σ > 0).
32
creasing in the coupon rate κ, increasing in the GDP haircut post default 1− α, increasing
in the haircut parameter θ, decreasing in the expected autarky time 1/λ.
One particular comparative static result worth highlighting is the fact that the default
boundary (as well as the country’s welfare) is decreasing in the coupon rate κ. This will have
its importance when I analyze the Latin American debt crisis of the early 1980s – at that time,
most of the sovereign debt contracted by these small open economies was structured with
variable coupons indexed to US short term rates. As will be seen in Section 7, the increase
in short term rates creates a current account reversal that is magnified by the contractual
structure of sovereign debt at the time.
5 Applications
5.1 Numerical Illustration and Comparative Static Results
Before estimating my model for a set of countries of interest, I first provide some comparative
static analysis, in order to gain some intuition about the role of certain model parameters.
To facilitate this investigation, I shut down for now the multiple SDF regimes, and analyze
the effect of certain model parameters in an environment where there is only one SDF regime.
I solve the model numerically using a Markov chain approximation method, as described in
details in Section A.3.
5.1.1 Calibration
Table 1 highlights the base case parameters I use for this comparative static analysis. I select
model parameters that are meant to represent the “average” emerging market economy of
the dataset studied in the online appendix. More specifically, the table “Country-Specific
Macro Moments” in the online appendix, constructed using data from the World Bank, shows
that the average real GDP growth rate for the countries in my dataset is 3.5% p.a., and the
average real GDP growth volatility is 4.1% p.a., leading to the parameters µ and σ in Table 1.
In the same online appendix, the table “Bond Issuance Average Maturities” shows summary
statistics for a dataset of foreign currency sovereign bonds I collected from Bloomberg13;
according to such table, the average original maturity date of sovereign bonds issued by my
13For all countries in the data-base I construct, I download all bonds listed on Bloomberg and issued bysuch country. I only keep in my data-base foreign currency bonds denominated in either EUR, GBP, USD,JPY or DEM. I also exclude bonds whose original notional amount is less than USD 100mm, whose originalterm is less than 1 year or greater than 50 years, or bonds with non-fixed coupon rates. The list of remainingbonds is available upon request.
33
Parameter Value Description1/ρ 0.5 IESγ 5 Risk aversionδ 0.2 Rate of time preferenceµ 0.035 GDP growth rate|σ| 0.04 GDP growth volatility
1− α 0.04 GDP % fall at default1/λ 5 Capital markets’ exclusion (years)θ 0.50 Debt-to-GDP upon autarky exitr 0.05 Creditors’ risk free rate|ν| 0.625 Creditors’ market price of risk
corr(ν,σ) 0.50 Business cycle-risk price correlation1/m 7 Debt average life (years)κ 0.05 Debt coupon rate
Table 1: Calibration Parameters
sample of 27 emerging market countries is 13.8 years. Since a country consistently rolling over
13.8-year original maturity bonds has a debt average life of 6.9 years, I pick 1/m = 7 years.
I select jointly the default punishment parameter 1− α and the rate of time preference δ to
approximately match two moments of the data: the average debt-to-GDP ratio (equal to 50%
for the set of countries of focus, as indicated in the “Country-Specific Macro Moments” table
in the online appendix), and the average 5yr CDS premium (equal to 395bps for the set of
countries of focus, as indicated in “Country-Specific Debt Price Moments” table in the online
appendix). This procedure yields a permanent output drop upon default of 1 − α = 4%14,
and sovereign rate of time preference δ = 20%. According to Tomz and Wright (2013), it
takes 4.7 years post-default for a country to regain access to capital markets, leading me to
pick 1/λ = 5 years. Benjamin and Wright (2009) find a mean creditor haircut following a
sovereign default of approximately 40%. Since my model assumes that the face value of debt
at exit from financial autarky is equal to θYτ+τe/Yτ− times the face value of debt pre-default,
the model-implied face value haircut is equal to:
1− θαE[e(µ−
12σ2)τe
]= 1− αθλ
λ−(µ− σ2
2
)14Note that estimates of output drops following a sovereign default vary vastly across the empirical litera-
ture: Hebert and Schreger (2016) for example calculate the cost of Argentina’s sovereign default to correspondto 9.4% permanent reduction in output; Aguiar and Gopinath (2006) use a (transitory) cost of default of 2%of output in their model, citing evidence from Rose (2005), who calculates an 8% decline in internationaltrade following a sovereign default.
34
I thus pick θ = 0.50, leading to a model-implied average creditor haircut of 42%. I choose
an inter-temporal elasticity of substitution equal to 1/ρ = 0.5 that is consistent with the
international business cycle literature15, and a risk-aversion parameter γ = 5 that is consistent
with the asset pricing literature. Risk-free rates are set at 5% p.a., and I pick a sovereign debt
coupon rate of 5%, which means that (a) the risk-free value of government debt is 1, and (b)
the sovereign debt always trades at a discount to par. The magnitude of the risk price ν is
set at the ergodic mean risk price of Lettau and Wachter (2007) (the pricing kernel I will use
later on when incorporating multiple SDF states). In a model with one risk price state (and
therefore no SDF jumps), |ν| corresponds to the highest Sharpe ratio attainable by any asset
priced by international investors (see Hansen and Jagannathan (1990))16. To start with, I
assume a correlation between the creditors’ risk price and the government’s output process
of 50%, and will discuss how this correlation affects equilibrium outcomes in Section 5.1.3
(of course in this simple model, varying this correlation and keeping risk prices constant is
equivalent to keeping this correlation constant and varying risk prices). In all the figures I
will be discussing, I plot in dotted lines the ergodic distribution of the state variable xt in
order to focus my attention on the sub-interval of the state space where I expect to see most
of my model-implied observations. This ergodic distribution is obtained via integrating a set
of Kolmogorov forward equations, as Section A.1.14 reveals. In the case of a unique SDF
regime, the ergodic distribution admits the following semi-closed form expression:
Lemma 4. In the absence of multiple SDF states, the ergodic distibution f takes the following
form:
f(x) =
∫ xθx
exp[∫ t
x2
|σ|2s2 ((m+ µ)s− ι(s)) ds]
2G|σ|2t2dt if x ∈ [0, θx)∫ x
xexp
[∫ tx
2|σ|2s2 ((m+ µ)s− ι(s)) ds
]2G|σ|2t2dt if x ∈ (θx, x)
The constant G is pinned down by the condition∫ x
0f(x)dx = 1− 1
1+λT (θx), where T (x) is the
expected default time conditional on the initial debt-to-GDP ratio being equal to x.
5.1.2 Base Case Calibration Equilibrium
In Table 2, I display the key model-implied moments of interest, as well as the target and
historical values for the “average” of my 27 countries in the online appendix. The base case
15Aguiar et al. (2016b), Aguiar and Gopinath (2006), Arellano (2008) all use an IES of 0.5.16Note that this 62.5% maximal Sharpe ratio is slightly higher than the unconditional Sharpe ratio of 40%
obtained for US equities – computed using an historical average of US equity market excess returns of 6%,and a yearly volatility of 15%.
35
Target/Moment Notation Value Historial Valuedefault boundary x 56%mean debt-to-GDP E [xt] 52% 50%stdev debt-to-GDP stdev [xt] 3% 24%consumption-output vol ratio vol (Ct) /vol (Yt) 1.94
default rate (1/λ+ T (θx))−1 2.8% p.a.bond spread E [ς(xt)] 365 bps p.a.5y CDS spread E [ς(xt, 5)] 414 bps p.a. 407 bps p.a.5y-1y CDS slope E [ς(xt, 5)− ς(xt, 1)] 60 bps p.a. 72bps p.a.bond excess return E [dRe
t ] 164 bps p.a.bond return volatility stdev [dRe
t ] 525 bps p.a.5y CDS excess return E
[dRe
t,5
]230 bps p.a. 508 bps p.a.
5y CDS return volatility stdev[dRe
t,5
]732 bps p.a.
Table 2: Base Case Calibration Results
calibration results in an optimal debt-to-GDP default boundary x = 56%, an ergodic mean
debt-to-GDP ratio of 52% and an ergodic debt-to-GDP distribution standard deviation of
3.0%. In Figure 2, I plot the issuance policy ι and the resulting trade balance. The issuance
policy is a decreasing function of the debt-to-GDP ratio, positive but reaching levels close
to zero at the default boundary. It is important to keep in mind that these represent gross
issuances, before taking into account any debt amortization. In Figure 2a, the dotted blue line
represents the locus of points (µ+m− |σ|2)x, i.e. the required value of ι(x) such that the
drift rate of x is zero. It is immediate to notice that the debt-to-GDP ratio is a mean-reverting
variable – its drift rate is going to be positive for values of x on the left of the intersection
of the solid and dotted blue lines and negative on the right side of such intersection. The
resulting trade balance (as a fraction of GDP) is equal to 1 − c(x) = (κ + m)x − ι(x)D(x).
It is negative for low debt-to-GDP levels and positive otherwise. Thus, consistent with
the overwhelming data for emerging market economies, the trade balance in this model is
countercyclical.
Since the consumption-to-output ratio c(x) is a decreasing function of the debt-to-GDP
ratio, using equation (34) it is immediate to see that the model generates a consumption
growth volatility that is greater than output growth volatility, an empirical regularity of the
data. For the parameters selected, I obtain an ergodic consumption growth vol to output
growth vol ratio of 1.94, which is in line with several emerging market economies, as docu-
mented in Aguiar and Gopinath (2004)17. Note also that consumption growth and output
17They find a ratio of 1.38 for Argentina, 2.01 for Brazil, 2.39 for Ecuador, 1.70 for Malaysia, 1.24 forMexico, 0.92 for Peru, 0.62 for the Philippines, 1.61 for South Africa, 1.09 for Thailand and 1.09 for Turkey.
36
Figure 2: Government Financing Policy and the Trade Balance
growth are perfectly correlated (at least locally) in a model with one SDF regime, which is
obviously counter-factual – the model featuring multiple SDF regimes breaks this result.
I plot the sovereign bond price and the sovereign spread in Figure 3. The ergodic mean
credit spread of the exponentially amortizing bond is equal to 365bps p.a., and the ergodic
credit spread volatility is equal to 132bps p.a. This ergodic mean credit spread volatility is
higher than what is obtained by Aguiar et al. (2016b) and the difference stems from the much
lower average maturity of sovereign debt (2 years) this latter article assumes, compared to
the 7-year average life debt in the data for my countries of interest.
Focusing on Figure 3, bond prices decrease with the debt-to-GDP ratio, while bond
spreads increase. Even at low debt-to-GDP ratios (i.e. lower than the ergodic debt-to-GDP
mean), sovereign spreads are far from negligible. When the sovereign has no debt outstanding,
sovereign spreads are strictly positive – a simple manipulation of equation (20) shows that
the sovereign spread at x = 0 verifies:
ς(0) = −ι(0)D′(0)
D(0)
This equation highlights the role of future debt issuances (and the implicit dilution risk
associated with those future debt issuances): at low debt-to-GDP ratios, creditors perfectly
anticipate that the government will be issuing large amounts of debt (since the government
37
Figure 3: Government Bond Prices and Spreads
(a) Bond Prices D(x) (b) Bond Spreads ς(x)
is impatient), leading them to price bonds at a discount that reflects such dilution risk.
I then focus on credit default swap premia ς(x, T ). While the government uses 7-year
bonds to smooth and front-load consumption, my model allows me to compute the premium
ς(x, T ) of CDS contracts at any time-horizon T , as discussed in Section 3.6. The numerical
procedure to solve the relevant PDEs is described in details at the end of Section A.3. I plot
CDS premia in Figure 4a for 1-year, 3-year and 5-year contracts, and I plot in Figure 4b the
5y-1y slope. CDS premia for short-dated contracts are close to zero when the debt-to-GDP
ratio is far away from the mean ergodic debt-to-GDP ratio, highlighting the fact that at such
low debt-to-GDP levels, the sovereign slowly increases its indebtedness, such that 1-yr credit
instruments are almost risk-free. The 5-year CDS premium ς(x, 5) resembles the credit spread
of the exponentially amortizing bond ς(x) used by the government to finance itself, since the
average life of this bond is 7 years. Lastly, the term structure of credit spreads becomes
inverted as the sovereign approaches its default boundary. This property of my model ends
up being a very general property of default models that are structured as “first-hitting-time”
models, where the state is mean-reverting and has continuous sample paths. This feature of
the model also fits with stylized fact (5).
I plot in Figure 5 the bond expected returns and the bond price elasticity. Expected excess
returns are non-zero, since I have used a pricing kernel that co-moves with the country’s
output process: the unconditional expected excess return is equal to 1.64% p.a., while the
38
Figure 4: Credit Default Swap Premia
(a) CDS ς(x, T ) (b) CDS Slope ς(x, 5)− ς(x, 1)
Figure 5: Expected Excess Returns and Bond Price Elasticity
(a) Expected Excess Returns (b) Bond Price Elasticity −xD′(x)
D(x)
39
unconditional return volatility is equal to 5.25% p.a. Expected excess returns increase with
the debt-to-GDP ratio, since the bond price function’s elasticity increases with the state
variable x – this is the model counterpart to fact (8) in the data. I plot the bond price
elasticity −xD′(x)D(x)
in order to get a better understanding of the magnitude of the sovereign
bond’s US equity market beta – this will help me understand whether my model stands a
chance at generating the magnitude of expected excess returns observed in the data. In the
one-state SDF model, conditional expected excess returns are equal to −xD′(x)
D(x)σ ·ν. The risk
prices will be time-varying in the full model, but will on average be equal to 62.5%. Most
of my countries of interest have GDP volatilities of the order of 4%, meaning that at best,
an emerging market output’s risk premium is equal to σ · ν ≈ 2.4%. In this one-state SDF
model, the bond price elasticity has an ergodic average equal to 1.33. Thus, at best (in other
words when corr (ν,σ) = 1), the ergodic mean excess return will be equal to 3.3% – high,
but not quite sufficient to reach the unconditional expected excess returns in the data for
certain countries such as Brazil or Hungary for example.
I end this section by focusing on the historical measure. When the country emerges
from financial autarky, it has a debt-to-GDP ratio θx = 28.5%, and it takes such country
on average 24 years to default on its debt, once it has exited from autarky. The sovereign
default frequency under the physical measure is thus equal to 2.8% p.a. – slightly higher
than the unconditional estimate of 2% cited in multiple studies, but this measure is of course
extremely difficult to estimate accurately in the data given the low frequency nature of
sovereign defaults18.
After having investigated some of the key outputs of the model in the base case parametriza-
tion, I now study the impact of the model parameters on various endogenous quantities of
interest. This comparative static analysis will be used in my estimation in order to provide
identification for a subset of parameters of my model.
5.1.3 Comparative Statics
Table 3 is a summary of the comparative statics with respect to several parameters of the
model. These comparative statics are performed starting from the base case parametriza-
tion of Table 1. In the table, I compute the elasticity of the moment of interest w.r.t. the
parameter of interest; blue numbers correspond to positive elasticities, red numbers corre-
18It is also worthwhile noting that this historical default frequency is high due to the assumption that theloss severity suffered by bond investors in connection with a sovereign default is not 100%. In the discrete timeliterature on sovereign defaults, barring a few exceptions, most papers assume that creditors’ loss severityupon a sovereign default is 100%. In the case of creditors’ risk-neutrality, this automatically causes the levelof credit spreads to be close to the level of historical default intensities; if instead a recovery R is realizedby creditors, the historical default intensity would be approximately 1/(1 − R) times the average sovereigncredit spread.
40
spond to negative elasticities. As an example, if one focus on the ergodic debt-to-GDP mean
and the variation of such mean w.r.t. the impatience parameter δ, the elasticity is equal
to ∂ lnE[xt]∂ ln δ
= −0.56, meaning that a 10% increase in the value of the parameter δ leads to
5.2% decrease in the ergodic mean debt-to-GDP ratio. These elasticities are closely related
to the covariance matrix of parameter estimates that will be calculated as part of my model
estimation.
The GDP drop upon default 1 − α has a significant impact on government behavior –
specifically on the debt-to-GDP ratio at default, its ergodic mean and standard deviation.
An increase in the default punishment incentivizes the government to support higher levels
of debt in equilibrium, a result already known in the literature using discrete time models.
The magnitude of the sensitivities are not surprising: my output process being a geometric
Brownian motion, any downward GDP drop is a permanent shock that is translated into
large welfare losses. Since the default boundary increases with the magnitude of the GDP
drop post-default, the government issuance policy adjusts upwards, via an almost-parallel
shift.
As the government becomes more impatient (i.e. as δ increases), it tends to front-load
consumption when it is not significantly indebted. The impatient government thus has a
debt issuance policy with a steeper slope than the patient government, and the optimal
default boundary is lower for the former than for the latter. With an impatient government,
creditors take into account the dilution risk and price the debt more punitively than in the
situation where the government is patient, which leads to higher credit spreads, and a much
steeper 5yr-1yr spread slope. The long-run mean spread volatility is higher, which leads to
significantly higher risk-premia.
My framework allows me to investigate the separate roles of risk-aversion and inter-
temporal elasticity of substitution in the government’s decision problem – an analysis that the
international macroeconomic literature has not focused on so far. An increase in risk-aversion
tends to decrease the debt issuance rate, due to a greater precautionary savings motive.
It also increases the incentive to default, since welfare costs of business cycle fluctuations
increase with the level of risk-aversion, and since consumption volatility is greater than
output volatility in this class of models. Thus, the equilibrium default boundary is lower
with a more risk-averse government.
Increases in GDP volatility decrease the equilibrium debt-to-GDP default boundary. Once
again, this is due to the increase in the incentive to default, since welfare costs of business
cycle fluctuations increase with the GDP volatility, and since consumption volatility is greater
than output volatility. The mean bond spread increases significantly, since if one keeps the
government financing policy unchanged, a greater GDP volatility increases the default barrier
41
hitting probabilities (of course there will be an equilibrium response by the government, which
will reduce its debt issuances). A greater GDP volatility increases bond spread volatilities,
and as expected the sovereign bond risk-premium is materially higher. More surprisingly, a
greater GDP volatility does not lead to a meaningfully wider debt-to-GDP distribution, as
illustrated by the low sensitivity of stdev [xt] to |σ|. This is disappointing since it suggests
that stochastic volatility will not rescue one fundamental weakness of this class of models: the
fact that the model-implied ergodic debt-to-GDP distribution is a lot thinner than the one
observed in the data – the latter being computed and displayed in table “Country-Specific
Macro Moments” in the online appendix.
5.1.4 Breaking Down Governments’ Cost of Financing
In order to better understand the impact that SDF regime shifts will have on equilibrium
outcomes, I look at the comparative static w.r.t. ν in more details. Higher risk prices in-
crease sovereign spreads, lower government bond prices, leading the government to adjust
its issuance policy downwards. Higher creditors’ risk prices also cause the sovereign govern-
ment to default at lower debt-to-GDP levels – although the adjustment is relatively small.
This mechanism leads, in a multi-SDF-regime version of my model, to jump-to-default risk,
induced by risk prices jumping from one level to another (higher) level. When comparing
credit spreads for ν = 0 to credit spreads with strictly positive risk prices, one might want to
interpret the spread differential between those two parameter configurations as the “spread
premium” that a sovereign government is paying to its creditors above and beyond what
would be actuarially fair. This is not exactly the case since the government endogenously
reacts to those higher risk prices by altering its issuance and default policies.
I thus look at the following credit spread decomposition: starting from the equilibrium
without risk-pricing, I first adjust risk prices to |ν| = 0.625 (and under the assumption thatν·σ|ν||σ| = 0.50) and re-compute sovereign debt prices and spreads, keeping the government’s
issuance policy constant. I then adjust the government’s issuance policy to reflect such
government’s “best response” to this new debt price schedule. I will use this decomposition
in Section 6 in order to compute the cost borne by governments when issuing bonds to risk-
averse (US-based) investors. As discussed, this cost cannot be computed by simply evaluating
the average excess return paid to creditors for taking on sovereign credit risk, given the fact
that sovereign issuance policies depend on investors’ risk attributes; a counterfactual analysis
is required to decompose this sovereign financing cost.
The result of such decomposition is illustrated in Figure 6: the credit spread function
plotted in red is the spread under the intermediate step where risk prices are set to 0.625
but where the government issuance policy has not reacted yet. Figure 6 shows that positive
42
Figure 6: Credit Spread Decomposition
risk prices alone act as a powerful force to push credit spreads wider. Indeed, when |ν| is
increased from zero to |ν| = 0.625 (and the issuance policy is not adjusted), the drift rate of
the state variable xt increases by xtσ ·ν ≈ 0.5×0.04×0.625×0.5 = 0.625% per annum (using
the ergodic debt-to-GDP ratio for xt and the assumed 50% correlation between risk prices
and GDP), and the credit spread is wider by 233bps p.a. on average (this can be seen in the
plot by the upward shift from the curve ν = 0 to the curve ν = 0.625∗). But the government
responds by adjusting its issuance policy downwards, defaulting at a lower debt-to-GDP
burden, and the resulting credit spreads ends up wider by “only” 101bps p.a. on average
(using the ergodic distribution with risk-pricing). In other words, in this simple calibration,
101/360 ≈ 28% of total financing cost of the government is attributable to investors’ risk
aversion. For my panel of countries of interest, as will be seen shortly, this cost will turn out
to be very close to 30%.
6 Structural Estimation
6.1 Procedure
I select a subset of N = 8 countries out of the set of 27 countries discussed in the online
appendix and for which I have the longest time-series data available. My subset consists
of Brazil, Bulgaria, Hungary, Indonesia, Mexico, Philippines, South Africa and Turkey. My
43
model with SDF regime shifts has a large number of parameters to determine. I am going
to impose restrictions on those parameters as follows. First, I will use a pricing kernel
specification widely used in the asset pricing literature to rationalize properties of US equity
market returns: the pricing kernel of Lettau and Wachter (2007), originally built to explain
(amongst other things) the value premium. This SDF features a constant risk-free rate (2%
per annum), and risk prices that follow an AR(1) process. Section A.4 provides a detailed
description of the properties of this SDF, as well as the method I use to transform the original
continuous-state specification of Lettau and Wachter (2007) into a discrete state continuous-
time Markov process νstt≥0, using a numerical procedure based on matching conditional
and unconditional moments of the original model and the approximating model. This gives
me the risk free rates rii≤Ns , risk prices νii≤Ns , the matrix of intensities Λ as well as the
SDF jumps υ(i, j)i,j≤Ns . I choose a number of states Ns = 5 in order to be able to solve
for a single equilibrium of my model in a few minutes of computing time. My risk prices
νii≤Ns are equally spaced between 0% and 150%, with an ergodic mean of 62.5%.
The remaining parameters to estimate are country-specific. For simplicity and due to the
small number of GDP data points available, I will assume that for each country of interest,
expected consumption growth, consumption growth volatility and the correlation between
the country’s output process and the SDF risk price do not change with SDF regime shifts,
in other words for each country n, µni = µnj , σni = σnj andσni ·νi
|σni ||νi|
=σnj ·νj
|σnj ||νj |
for any pair of SDF
states i, j. I will also assume that the sovereign government has time-separable preferences,
in other words I will assume that γn = ρn for each country n. A few other parameters are
calibrated using a-priori evidence. I will leverage the average original maturity of bonds
issued by each country (as documented in the “Bond Issuance Average Maturities” table
in the online appendix) in order to calibrate the debt parameter mn for country n19. The
expected time spent in financial autarky 1/λ and the parameter θ governing the debt-to-GDP
post-autarky are kept at their values in Table 1 and are thus not country-dependent. For each
country n, my estimation will then pin down the GDP growth µn, GDP volatility |σn|, the
preference parameter γn = ρn, the correlation between GDP and the risk price vector ν·σn
|ν||σn| ,
the GDP drop upon default 1− αn and the rate of impatience δn. The following 6 moments
will be used in my estimation. First, the first difference mean and standard deviation of log
output will provide information on µn and |σn|. Unconditional expected excess returns on 5y
CDS contracts will then provide information on the correlation between country n’s output
and risk prices. Finally, the level of 5y CDS premia, the mean debt-to-GDP ratio and the
term structure slope (i.e. the difference between 5y CDS and 1y CDS spreads) will jointly
19A country consistently rolling over T -year original maturity bonds has a debt average life of T/2 years;for such country I thus use λ = 2/T .
44
provide information on 1− αn, δn and γn.
My simulated method of moment estimation follows closely Lee and Ingram (1991).
I note H = 1T
∑Tt=1 ht the p × 1 vector of target moments in the data, and Hk(Θ) =
1T
∑Tt=1 h (xnt n≤N , st; Θ) the corresponding p × 1 vector of moments generated by the kth
simulation of my model. I use K = 1000 simulations, and note H(Θ) = 1K
∑Kk=1Hk(Θ)
the sample average of moments generated by my model across the K simulations. For each
simulation, I use an identical seed and generate 2T years of data, burn-in the first T years,
keeping only the simulated data for the last T years. I minimize the criterion function:
Θ = arg minΘ
(H −H(Θ))′W (H −H(Θ))
I use the diagonal matrix W = (diagi (H2i ))−1
as my weighting matrix, to penalize propor-
tional deviations of the model-implied moments from their data counterparts. I compute the
asymptotic covariance matrix of my estimator as follows:
covar(
Θ)
=1
T
(1 +
1
K
)(∂H∂Θ
′W∂H∂Θ
)−1∂H∂Θ
′W ′ΩW
∂H∂Θ
(∂H∂Θ
′W∂H∂Θ
)−1
In the above, Ω is a consistent estimator of the long run covariance matrix of the moment
conditions Ω =∑+∞
j=−∞ E[(ht − E [ht]) (ht−j − E [ht−j])
′]. Since my moments of focus are
computed using different data sets, over different time periods and with different frequencies
of data available, computing a consistent estimator of Ω (using Newey and West (1986) for
example) is difficult. Thus, instead, I compute an estimator of Ω using the null of my model:
Ω(Θ) :=1
K
K∑k=1
(Hk (Θ)−H (Θ)) (Hk (Θ)−H (Θ))′
Under the null hypothesis, Ω(Θ) is a consistent estimator of Ω.
6.2 Results – the Case of Brazil
In Table 4 and Table 5, I first display the parameter estimates, then the target and model-
implied moments, and finally additional moments that were not targeted in my estimation.
I display in figures 7, 8 and 9 the equilibrium outcome for Brazil. In the highest risk price
state, Brazil defaults at a debt-to-GDP ratio of 34.8%, whereas in the lowest risk price state,
it defaults at a debt-to-GDP ratio of 36.3%, illustrating the relatively small impact of risk
prices on optimal default boundaries.
Figure 7 highlights the endogenous response of Brazil’s financing policy, trade and current
45
account balance to changes in international financial market conditions. Upon an increase
in the price of risk, its bonds trade at steeper discounts, inducing Brazil to reduce its debt
issuances. Jumps from the lowest risk price state to the highest risk price state lead to a large
adjustment to the current account and trade balances of approximately 4% of GDP. Thus,
increases in international risk prices constitute an endogenous mechanism for generating
sudden stops. Across my model economies, the adjustment in the trade and current account
balances varies beween 3% and 5%. While this is not sufficient to explain the full adjustments
observed in South East Asia post 1997 in connection with the Asian Tiger crisis (as illustrated
in Figure 16), those adjustments are nonetheless material. The mechanism used to generate
endogenous sudden stops in this paper thus differs from the more standard channel that has
been investigated in the past by the international macroeconomics literature – occasionally-
binding borrowing constraints in RBC models, such as those featured in Mendoza (2002) or
Mendoza (2010), and which depress investment and output when the small open economy
suffers a sequence of bad fundamental shocks and bounces against such constraint.
Figure 8 then shows that 5yr CDS levels jump up with transitions from low risk price
states to high risk price states – at the mean of the debt-to-GDP ratio ergodic distribution
for example, 5yr CDS jumps by more than 300bps between the lowest and the highest risk
price states. The term structure of credit spreads may invert for two reasons according to
Figure 8b: either following a sequence of bad fundamental shocks that push Brazil’s debt-
to-GDP ratio closer to the relevant default boundary, or upon an upward shift in risk prices.
This feature of the model is thus consistent with the data, as suggested by facts (5) and (6).
Figure 9 illustrates aspects of my model that are consistent with facts (7) and (8). Indeed,
one can see that CDS expected excess returns increase with the debt-to-GDP ratio, which
was expected given that the elasticity of the bond price function is increasing in the debt-
to-GDP ratio (this is fact (7)). In addition, longer-term CDS contracts earn higher excess
returns than shorter-term CDS contracts, for a given debt-to-GDP level and a given risk
price level (this is fact (8)). This effect is due to the risk exposure of CDS contracts, which
is increasing in the tenor of such contract.
Table 4 and Table 5 also indicate the estimated average bond spread (for the exponentially
amortizing long term bond issued by each small open economy), as well as the average bond
spread that would prevail if investors were instead risk-neutral (in row titled “Bond spread
– RN model”). The difference between those two spreads can thus be viewed as the excess
compensation paid to sovereign debt investors for taking on risks that are correlated with
those investors’ marginal utility process. On average, approximately 30% of the credit spread
paid by sovereign governments above and beyond the risk free rate is attributable to investors’
As Table 4 and Table 5 indicate, my estimation leads to target model-implied moments
that are relatively close to the data, except for the 5y CDS expected excess return. That
moment turns out to be particularly difficult to hit: indeed, for most small open economies
in my sample, the estimated correlation between the GDP process and the risk price vector
has to be 100%, and model-implied bond and CDS expected excess returns are below those
in the data. High levels of correlations between the risk price vector and the GDP process
of small open economies also lead to counterfactually high cross-country GDP correlations,
another negative consequence. In my model, either risk prices or risk exposures are too low.
Whereas risk exposures are functions of the state variables and depend on the deep model
parameters, risk prices are taken from the SDF of Lettau and Wachter (2007), using the
observation that the CAPM is not rejected (meaning that US investors seem to be marginal
in sovereign credit markets). One way to address the issue of low model-implied expected
excess returns would be to add, as part of the estimation, all the SDF parameters. This
turns out to be a numerical challenge, which I plan on tackling in subsequent research.
In addition, the debt-to-GDP ergodic distributions generated by the model have much
smaller variances than in the data, even in the presence of time-varying risk prices. The
small variance of the model-implied debt-to-GDP ergodic distribution is mostly due to the
low volatility (in absolute terms) of GDP shocks suffered by my small open economies of
48
focus20. One might have conjectured that time-varying risk prices, and thus regime spe-
cific default boundaries, will increase the variance of the ergodic debt-to-GDP distribution.
Unfortunately, the optimal default boundaries do not strongly react to the presence of regime-
specific risk prices, since the government adjusts its issuance policy to reflect different credit
market conditions. Making the sovereign output volatility σt time varying would help slightly,
as suggested by some (unreported) experimentation I ran, but would in no way enable the
model-implied distribution variances to match those of the data: as table “Country-Specific
Macro Moments” in the online appendix indicates, the standard deviation of the empirical
ergodic debt-to-GDP distribution for my countries of focus is approximately 25%, which is
an order of magnitude larger than those obtained in my model.
7 The 1980’s Latin American Debt Crisis
In this section, I illustrate the flexibility of my framework by studying the 1980’s Latin
American debt crisis. For a variety of reasons documented in multiple historical studies (see
for example Dooley (1994) or Cline et al. (1995)) Latin American governments borrowed
heavily during the 1970s, partly as a consequence of an increase in the supply of loans from
US banks recycling petro-dollars, partly to take advantage of historically low real interest
rates in the US, and partly as a consequence of the need to finance large current account
deficits following the two oil price shocks of 1973 and 1979. Latin American sovereign debt
increased by an average of 24% per annum between 1970 and 1979, therefore substantially
increasing those countries’ debt-to-GDP ratios. The largest sovereign borrowers during that
time period were Mexico and Brazil. The World Bank estimates that two third of that
debt was in the form of USD-denominated, long-term, syndicated bank loans whose interest
rate was indexed to LIBOR, thereby making sovereign governments’ financing costs directly
exposed to the US dollar and US monetary policy. In the early 1980s, the Federal Reserve
aggressively increased US short term rates to fight domestic inflation, causing the US dollar to
appreciate against most currencies, and causing LIBOR rates to skyrocket. In August 1982,
as T-bill rates were approaching 16%, Mexico announced that it could no longer meet its
debt service payments; by the end of that year, 40 other nations, including Brazil, Venezuela
and Argentina, had defaulted on their sovereign debt.
My model allows me to perform a “lab experiment” on this historical period. Indeed, it is
straightforward to change the debt contract structure that the sovereign government enters
20While the GDP volatilities of my small open economies of focus are larger than those of developedeconomies, the absolute level of such volatilities has a first order impact on the variance of the ergodicdebt-to-GDP distribution.
49
into from fixed rate contracts to floating rate contracts. Thus, let me now assume that the
sovereign issues floating rate debt indexed to the risk free rate rst , which evolves according
to a discrete state Markov process with generator matrix Λ. In other words, coupon rates
paid by the government are time varying, and equal to κst = rst in state st. The government
resource constraint (out of financial autarky) becomes:
Ct = Yt + ItDt − (κst +m)Ft
Debt prices and the life-time utility function for the government satisfy second order ordinary
differential equations similar to those presented in Section 3.5, but appropriately modified to
account for the floating rate nature of sovereign debt. Default optimality is still obtained by
a set of smooth-pasting conditions. It is worthwhile noting that the price of a risk-free debt
instrument that amortizes exponentially and pays a coupon of rstdt for t ∈ [t + dt] is equal
to:
E[∫ ∞
0
e−∫ t0 (rsu+m)du(rst +m)dt|Ft
]= 1
In other words, the price of risk-free floating rate debt in this set up is always par, and any
credit-risky floating rate instrument (where the coupon paid is equal to the floating rate
benchmark) will trade at a discount to par.
In order to compute US short term real risk-free rates, I proceed as follows. I download
the US one-month T-bill rate from Ken French website; this will be my time series of US
nominal short term risk free rates. I download the consumer price index (CPIAUCSL) from
the Saint Louis Fed website, and compute expected inflation by using the one-month ahead
inflation forecast generated from a univariate AR(5) of inflation estimated using the previous
10 years of inflation rates. I then subtract my measure of expected inflation from the nominal
rate in order to obtain a measure of real short term interest rates. The US nominal and real
short term interest rates obtained are plotted in Figure 13. I then use the time series of US
short term real rates in order to estimate the generator matrix Λ, after having discretized
the interest rate process into a 4-state Markov process taking values (0%, 2%, 4%, 6%)21. The
time series of US real short term interest rates as well as the discrete state Markov process
approximation are plotted in Figure 14. On average, st spends 35% of the time in the 0%
interest rate state, 49% of the time in the 2% interest rate state, 12% of the time in the 4%
interest rate state and 4% of the time in the 6% interest rate state.
21I restrict the real rate to be weakly positive in my numerical application since the long term bond issuedby the sovereign government is linked to such short rate; negative real interest rates in my model would leadthe small open economy to receive payments from creditors.
50
For this exercise, I use model parameters as in Table 1; while I assume that creditors’
short rates are stochastic, I assume no risk-premia, in other words νs = 0 and υ(s, s′) = 0
for all pair of states s, s′. I then solve numerically the model for two separate contractual
structures of sovereign debt: first, when the coupon rate on the debt is indexed to US risk
free rates (which is the relevant case for Latin American economies in the early 1980’s), and
second, when the coupon rate on the debt is constant and equal to 2% (i.e. approximately
equal to the ergodic risk free rate of 1.7%), in order to understand the importance of the
contractual structure of sovereign debt.
The result of this exercise is striking, as showed in Figure 10, where I plot the equilibrium
trade balance of my model small open economy when the debt issued has floating coupons
(Figure 10a) and fixed coupons (Figure 10b). With floating rate coupons, the debt-to-GDP
default boundaries differ significantly state by state, from 74% in the the lowest US short-
rate state to 70% in the highest US short-rate state. Instead, with fixed coupon debt, the
debt-to-GDP default boundaries are clustered around 74%. In addition, for both contractual
structures, issuance policies depend in a negative way on short rates: for a given debt-to-
GDP ratio, the lower the interest rate, the higher the issuance rate, leading to current account
adjustments when US short term interest rates increase. However, while those current account
adjustments are relatively small (of the order of 1%) in the case of fixed coupon debt, they
are materially larger when the small open economy issues floating coupon debt – at the
ergodic mean of the debt-to-GDP distribution, a jump from 0% US interest rates to 6%
US interest rates leads to a current account adjustment of approximately 7%. Of course,
if the small open economy’s debt-to-GDP ratio is materially above its ergodic mean at the
time of the interest rate increase, the government might be induced to “jump-to-default”.
Lastly, a US interest rate increase is experienced by all small open economies at the same
time, inducing correlated defaults and correlated current account adjustments for those small
open economies that financed themselves using floating coupon debt. The magnitude of
the current account adjustment obtained in the model with floating rate debt corresponds
approximately to those observed during the large expansion of foreign currency sovereign
debt experienced by Latin American economies in the 1970s, as well as following the Volcker
shock post-1982, as shown in Figure 15. The figure shows that the current account balance
for Argentina, Brazil, Colombia, Mexico and Peru was negative and between -1% and -8%
in the late 1970s and early 1980s, but adjusted upwards at the end 1982 by up to 7% for
Mexico. Thus, qualitatively, the mechanism highlighted in this section – a large rise of US
real rates combined with floating coupon sovereign debt – is a plausible explanation for the
behavior of the current account balance of these emerging market economies before and after
the Volcker shock.
51
Figure 10: Trade Balance 1− ci(x)
(a) Floating Rate Debt (b) Fixed Rate Debt
To understand in more details the mechanism driving the current account behavior dis-
cussed above, it is worth looking at the financing policy of the government when bonds that
are issued have floating coupons. The optimal Markov issuance policy has the following
expression:
ιi(x) =1
Di(x)
(δDi(x) [(1− γ)vi(x)]ρ−γ1−γ
−v′i(x)
)1/ρ
+ (κi +m)x− 1
The expression above indicates that for a given debt price schedule D and life-time utility
set of functions v, the issuance rate increases with the coupon rate paid on the debt – this
is a cash-flow effect, due to the fact that in a high interest rate environment, higher debt
servicing costs incentivize the government to issue larger amounts of debt to achieve the same
level of consumption than in a low rate environment.
So how can we explain the fact that the government issues more debt in a low rate
environment, and defaults at a higher debt-to-GDP level? As discussed previously, buyers
of risk-free floating rate debt are not exposed to the level of short rates – in other words,
the interest rate duration of their investment is zero. On the other side, the sovereign
government issuing such floating rate bonds is not indifferent: an increase in short term rates
makes the country strictly worse off, since its financing cost is higher. Thus, everything
52
Figure 11: Credit Spreads ςi(x)
(a) Floating Rate Debt (b) Fixed Rate Debt
else equal, one would expect a country to default at a lower debt-to-GDP level when short
term interest rates are higher. This means that the price of floating rate sovereign debt
should decrease when short term interest rates jump up, and that credit spreads should be
higher – this is effectively what happens, as illustrated in Figure 11. Since the credit cost of
issuing debt increases in a high interest rate environment, the sovereign government adjusts
it issuance strategy downwards. In addition, the value function for the government (as a
function of the debt-to-GDP ratio) is steeper in a high interest rate environment: indeed,
when the government has no debt, one would expect the value function of the government
to not be very sensitive to the level of short term rates; instead, when the government is
highly indebted, one would expect the value function to be significantly lower in a high rate
environment than in a low rate environment. In other words, |v′r=6%(x)| > |v′r=0%(x)|, i.e.
the marginal cost of issuing debt is higher in a high US interest rate environment, pushing
further down the issuance policy of the government, and exacerbating the current account
adjustment upon an increase in US short rates.
8 Endogenous Growth
The model studied so far exhibits countries whose output process is specified exogenously.
The government’s sole motive for taking on debt is a consumption tilting and a smoothing
53
motive. What happens when the government instead borrows from external creditors in order
to finance domestic investments and capital accumulation? This section answers this ques-
tion, and emphasizes two sources of debt overhang channels through which highly indebted
sovereign government’s capital investment decisions might be distorted. This enhanced model
provides a simple micro-foundation to the output process of Aguiar and Gopinath (2004),
Aguiar and Gopinath (2006), and many other articles in the sovereign debt literature, in
which output growth is a mean-reverting variable.
Assume now that the country n has a production technology Yt = aKt, with a > 0,
where Kt is the number of effective units of capital in the small open economy. Assume that
effective capital evolves according to:
dKt = (Ht − ηKt)dt+Ktσ · dBt (40)
Ht represents effective capital investments and η is the rate of depreciation of capital. The
effective capital in this economy is hit by capital quality shocks similar to those in standard
continuous time macro models such as Brunnermeier and Sannikov (2014) or He and Kr-
ishnamurthy (2014). Capital investments come with adjustment costs equal to G(Ht, Kt) =
g(Ht/Kt)Kt, with g strictly convex, g(0) = g′(0) = 0. For simplicity, I will assume that
g(h) = ψ2h2, where h is the investment rate per unit of effective capital. Government debt
dynamics continue to follow equation (7). The government’s resource constraint can now be
written:
Ct +Ht = aKt + ItDt − (κ+m)Ft −G(Ht, Kt) (41)
Equation (41) simply says that the sum of consumption Ct and investment Ht need to be
equal to the sum of output aKt and capital markets net flows ItDt − (κ + m)Ft, net of
investment adjustment costs G(Ht, Kt). If and when the government elects to default, its
(efficiency units of) capital stock suffers a discrete drop, falling from Kτ− to Kτ = αKτ−.
The government is also excluded from capital markets for an exponentially distributed time
period, and exits autarky with a debt-to-capital ratio that is θ times its debt-to-capital ratio
pre-default.
The strategy of the government now consists in choosing an issuance policy I, an invest-
ment policy H , and a default policy τ in order to maximize its objective function. The debt
price follows equations (12) and (13). I look for an equilibrium where the debt-to-capital ratio
xt := Ft/Kt and the SDF regime st are the state variables of this modified environment. The
details of the equilibrium calculations are displayed in Section A.1.19. When the debt-to-
capital ratio of the country is x and when the SDF state is i, the debt price is equal to Di(x)
and the life-time utility function for the government is equal to vi(x)K1−γ. The optimal debt
Tauchen, George and Robert Hussey. 1991. “Quadrature-based methods for obtaining ap-
proximate solutions to nonlinear asset pricing models.” Econometrica: Journal of the
Econometric Society :371–396.
Tomz, Michael and Mark LJ Wright. 2013. “Empirical research on sovereign debt and de-
fault.” Tech. rep., National Bureau of Economic Research.
Uribe, Martin and Vivian Z Yue. 2006. “Country spreads and emerging countries: Who
drives whom?” Journal of international Economics 69 (1):6–36.
61
A Appendix
A.1 Proofs and Detailed Calculations
A.1.1 A Useful Discrete Time Limit
In this section, I study a discrete time counterpart to the continuous-time model developed
in this paper. ∆t will represent a small time interval. I study the limit of a simple sovereign
default model as ∆t → 0, and show heuristically that no short term debt can be supported
in equilibrium at such limit. I focus on a government that has iso-elastic time-separable
preferences with rate of time preference δ and risk-aversion γ as follows:
E
[∞∑i=0
e−δi∆tC1−γi∆t
1− γ∆t|F0
]
Government output follows the discrete time equivalent of a geometric Brownian motion:
Y(i+1)∆t
Yi∆t= eω(i+1)∆t
In the above, ω(i+1)∆t is an F(i+1)∆t measurable binomial random variable that can take values
+σ√
∆t or −σ√
∆t with respective probabilities pu and pd:
pu :=1
2
(1 +
(µ− 1
2σ2)√
∆t
σ
); pd :=
1
2
(1−
(µ− 1
2σ2)√
∆t
σ
)
Thus, ω(i+1)∆ti≥0 is a sequence of i.i.d. random variables, and I will use ω∆t for simplicity
to denote one of these random variables. When ∆t→ 0, I have the following limits:
E [eω∆t ] = 1 + µ∆t+ o (∆t)
var [eω∆t ] = σ2∆t+ o (∆t)
The government has one-period debt at its disposal. Let Bi∆t be the stock of debt that
the government has to repay at time i∆t. Let D(B(i+1)∆t, Yi∆t
)be the price of one unit of
debt if the government plans to issue, at date i∆t, B(i+1)∆t units of debt maturing at period
(i+ 1)∆t. The government resource constraint at time i∆t is as follows:
Ci∆t∆t = Yi∆t∆t−Bi∆t +D(B(i+1)∆t, Yi∆t
)B(i+1)∆t
62
Upon default, GDP suffers a permanent shock of size 1 − α and the government is in au-
tarky forever after. Thus, if the pre-default output value is Yi∆t, the value function for the
government in default is equal to Vd(Yi∆t), which satisfies:
Vd(Y ) =(αY )1−γ
1− γ∆t+ e−δ∆tE [Vd (eω∆tY )]
Guessing that Vd(Y ) = vdY1−γ, the constant vd is equal to:
vd =
α1−γ
1−γ ∆t
1− e−δ∆tE [e(1−γ)ω∆t ]=
α1−γ
1−γ
δ + (γ − 1)(µ− 1
2γσ2) + o(1)
The latter equality is valid when ∆t→ 0. The government problem is as follows:
V0 (B, Y ) = max (Vc (B, Y ) , Vd (Y ))
Vd (Y ) = vdY1−γ
Vc (B, Y ) = maxB′
[(Y + 1
∆t(D (B′, Y )B′ −B)
)1−γ∆t
1− γ+ e−δ∆tE [V0 (B′, Y ′) |Y ]
]
The bond price verifies D(B′, Y ) = e−r∆t Pr (Vc (B′, Y ′) ≥ Vd(Y′)|Y ), where r is the interest
rate at which lenders discount risk-free cashflows. This formula assumes that upon default,
sovereign creditors recover nothing from their defaulted debt claim. One can show that for
any bond price function that is homogeneous of degree zero and decreasing in B′, the value
function Vc is homogeneous of degree 1−γ and the best “default” response by the government
is to follow a linear barrier policy of the form τ := inft : Bt ≥ xYt for some endogenously
determined constant x (which depends on the time step ∆t)22. Noting x := B/Y , and
using the homogeneity property of the value function, the government life-time utility can
be written Vc(B, Y ) = vc(x)Y 1−γ, and the government problem can be simplified as follows:
vc (x) = maxx′
[(1 + 1
∆t(D (x′)x′ − x)
)1−γ∆t
1− γ+ e−δ∆tE
[e(1−γ)ω∆t max
(vd, vc
(e−ω∆tx′
))]]
For simplicity, I assume that the choice set for the debt-to-GDP ratio of the government is
discrete. In other words, I discretize the state space into a grid G∆t consisting of a countable
22Of course this statement is assuming the existence of the value function Vc, which is the fixed point ofa functional equation. Since my discussion on the lack of equilibria with defaultable debt at the limit of mydiscrete time economies is only heuristic, I side-step the proof of existence of Vc.
63
number of points xi = eiσ√
∆t, for i ∈ N, where I note id the default index23. In other words,
I assume that at x = xid , the government strictly prefer defaulting over repaying, while
for i < id, the government weakly prefer repaying over defaulting. This model structure
guarantees that the debt-to-GDP ratio stays on the grid G∆t at all times, irrespective of the
sequence of GDP shocks and decisions made by the government. In any cutoff equilibrium
with default threshold x = xid , the debt price must satisfy:
D(xi) =
e−r∆t if i < id − 1
12
(1 +
(µ− 12σ2)√
∆t
σ
)e−r∆t if i = id − 1
0 if i ≥ id
Now consider the resource constraint for the government if at the beginning of a given period,
x = xi, with i = id − 1. Since i < id, it is optimal for the government to continue to
perform on its debt obligations. If the government selects a debt-to-GDP ratio xj < xi, the
consumption-to-output ratio, using the resource constraint, is equal to:
1 +1
∆t
(ejσ√
∆t−r∆t − eiσ√
∆t)
=(j − i)σ√
∆t+ o
(1√∆t
)→
∆t→0−∞
If the government selects a debt-to-GDP ratio xj = xi, the consumption-to-output ratio is
equal to:
1 +1
∆t
(1
2
(1 +
(µ− 1
2σ2)√
∆t
σ
)eiσ√
∆t−r∆t − eiσ√
∆t
)= − 1
2∆t+ o
(1
∆t
)→
∆t→0−∞
Of course if the government selects a debt-to-GDP ratio xj = xid , it obtains no proceeds from
its debt issuance – this decision cannot be optimal. Thus, as ∆t → 0, the only possibility
for government consumption to be positive at all points of the state space xi < x is for the
default boundary to converge to zero. Note that this conclusion stems from (A) the shape of
the government bond price schedule (at I take ∆t → 0, such debt price schedule converges
to a step function, equal to the risk-free benchmark for x < x and equal to zero at x ≥ x),
and (B) the continuous sample path of geometric Brownian motions. Had the output process
featured jumps, a cutoff equilibrium with strictly positive debt can be supported.
23This model structure implicitly assumes that the government is prevented from saving. This can also beachieved endogenously by using a sufficiently high rate of time preference δ.
64
A.1.2 Output Process in this Article vs. Aguiar and Gopinath (2006)
The output process in Aguiar and Gopinath (2006) can be recast in continuous time as
follows:
Yt = eztΓt
d ln Γt = ln gtdt
dzt = −κz (zt − µz) dt+ σzdBzt
d ln gt = −κg (ln gt − lnµg) dt+ σgdBgt
In the above, Bgt , B
zt are standard Brownian motions assumed to be independent, κz, κg
are positive constants that parametrize the speed of mean reversion of the processes zt and
ln gt respectively. Two approaches can be used to approximate the output process above by
equation (1).
In the first approach, conditional and unconditional moments of consumption growth can
be matched. One can for example match the s-lagged auto-correlation of log consumption
growth Rmodel(s) for both models. Such auto-correlations are equal to:
RAG(s) = e−κgs
RGBM(s) =
∑Nsi=1 πi
(µi − 1
2|σi|2
)∑Nsj=1
[(esΛ)ij− πj
] (µj − 1
2|σj|2
)∑Ns
i=1 πi(µi − 1
2|σi|2
)2 −(∑Ns
i=1 πi(µi − 1
2|σi|2
))2
One can also match the unconditional log consumption growth volatility Σmodel for both
models. Such log consumption growth volatilities are equal to:
ΣAG = σz
ΣGBM =
√√√√ Ns∑i=1
πi|σi|2
In the second approach, one can use a procedure introduced by Gordin (1969) to extract a
martingale component to the logarithm of output for both models, and match separately (a)
the volatility of the martingale component of both models, (b) conditional and unconditional
moments of the stationary component of both models, and (c) the time trend of both models.
For the model of Aguiar and Gopinath (2006), the decomposition of lnYt takes the following
65
form:
d lnYt =σgκgdBg
t − d(
1
κgln gt − zt
)+ lnµgdt
lnYt − lnY0 =
∫ t
0
σgκgdBg
s︸ ︷︷ ︸martingale component
+
[1
κgln g0 − z0
]−[
1
κgln gt − zt
]︸ ︷︷ ︸
stationary component
+ t lnµg︸ ︷︷ ︸time trend
Thus, the permanent component to log output is purely driven by (rescaled) shocks to the
state variable ln gt, while the state variable zt has a purely transitory role. The long-run
time trend in log-output is equal to lnµg. When I focus on the output process driven by
equation (1) (and assuming, for the purpose of this section, that Bt is unidimensional), I can
compute small increments in logarithms as follows:
d lnYt =
(µst −
1
2σ2st
)dt+ σstdBt
In the above, st is the discrete state Markov chain with generator matrix Λ and stationary
density π. In such case, note that:
E [d lnYt|Ft] =
(µst −
1
2σ2st
)dt
Let κ2,t :=∫ toσsudBu and note that for τ > t, E [κ2,τ |Ft] = κ2,t, in other words κ2,t is a
martingale. I then compute the long term average of log output growth:
ln g∞ : = π ·(µ− 1
2σ2
)I create the function f (s) := µs − 1
2σ2s − ln g∞ =
(µ− 1
2σ2 − ln g∞1
)· est , es is an Ns × 1
column vector with entry s equal to 1, and all other entries equal to zero, and compute:∫ ∞0
E [f (st+u) |Ft] du = Λ−1
(µ− 1
2σ2 − ln g∞1
)· est
In the above, Λ−1 is the generalized inverse of the generator matrix Λ (since the rows of the
matrix Λ sum up to zero, Λ is not invertible). Finally, I introduce the martingale κ1,t, defined
as follows:
κ1,t :=
∫ t
0
Ns∑i=1
Λ−1
(µ− 1
2σ2 − ln g∞1
)·(ei − esu−
)dN (su−,i)
u +
∫ t
0
(µ− 1
2σ2 − ln g∞1
)· esudu
66
I can then decompose increments in log output growth as follows:
d lnYt = d (κ1,t + κ2,t)− Λ−1
(µ− 1
2σ2 − ln g∞1
)·(est − est−
)dN
(st−,st)t + ln g∞dt
lnYt − lnY0 =
∫ t
0
d (κ1,s + κ2,s)︸ ︷︷ ︸martingale component
+ Λ−1
(µ− 1
2σ2 − ln g∞1
)· (es0 − est)︸ ︷︷ ︸
stationary component
+ t ln g∞︸ ︷︷ ︸time trend
Thus, in the case of this article, the martingale component of log output is the sum of a
Brownian process κ1,t and a jump process κ2,t, while the stationary component is a pure
jump process. By carefully parameterizing µ,σ,Λ, one can approximate the long run trend,
stationary and martingale components in Aguiar and Gopinath (2006) by a stochastic process
that follows equation (1).
A.1.3 No-Debt Life-Time Utility
I note Ji (Y ) the life-time utility value when the government cannot issue debt, when the
level of output is Y and when the Markov state is st = i. I postulate that this life-time utility
can be written Ji(Y ) = KiY1−γ for some constants Kii≤Ns to be determined. Ji verifies
the following HJB equation:
0 = ϕ (Yt, Jst(Yt)) +AJst(Yt)
For convenience, I have introduced the differential operator A, defined for any stochastic
process Ztt≥0 (belonging to an appropriate class of stochastic processes) as follows:
AZt := limε→0
E [Zt+ε|Ft]− Ztε
(43)
Injecting my guess function for Ji, I obtain the following equation, for i ≤ Ns:
0 = δ1− γ1− ρ
KiY1−γ
(1
[(1− γ)Ki]1−ρ1−γ− 1
)+ µi(1− γ)KiY
1−γ − 1
2γ(1− γ)|σi|2KiY
1−γ +Ns∑j=1
ΛijKjY1−γ
Dividing by KiY1−γ and simplifying, the set of constants K := Kii≤Ns must satisfy:[
diagi(Ai)−1− ρ1− γ
Λ
][(1− γ)K] = δ [(1− γ)K]
ρ−γ1−γ
67
Vector exponentiation in the equation above has to be understood element by element. Under
assumption 1, this set of Ns equations in Ns unknown has a unique solution.
A.1.4 Creditors’ Stochastic Discount Factor
The discussion below is substantially similar to the discussion in Chen (2010). Assume
a representative creditor whose equilibrium consumption Cc,t follows, under the physical
measure P:
dCc,tCc,t
= µc,stdt+ σc,st · dBt
In the above, stt≥0 follows a discrete state Markov process with generator matrix Λ =
(Λij)1≤i,j≤Ns . Ns is the total number of discrete states. Assume that such creditor ranks
consumption streams according to the utility specification:
Ut = E[∫ +∞
t
Ψ (Cs, Us) ds|Ft]
(44)
In the above, the aggregator Ψ is assumed to be equal to:
Ψ (C,U) := δc1− γc1− ρc
U
(C1−ρc
((1− γc)U)1−ρc1−γc
− 1
)
The representative creditor’s life-time utility function can be written Ut =(hstYt)
1−γ
1−γ , where
the constants hi1≤i≤Ns satisfy the set of non-linear equations:
0 = δc1− γc1− ρc
hρc−1i + (1− γc)µc,i −
1
2γc(1− γc)|σc,i|2 − δc
1− γc1− ρc
+Ns∑j=1
Λij
(hjhi
)1−γc
As Duffie and Skiadas (1994) shows, the marginal utility process associated with recursive
preferences can be written:
Mt = exp
[∫ t
0
∂Ψ
∂U(Cc,z, Uc,z) dz
]∂Ψ
∂C(Cc,t, Uc,t) (45)
Applying Ito’s lemma then leads to the following dynamic evolution of Mt:
dMt
Mt
= −rstdt− νst · dBt +∑st 6=st−
(eυ(st−,st) − 1
) (dN
(st−,st)t − Λst−,stdt
)(46)
68
In the above, N(st−,st)t is a Poisson counting process for transitions from state st− to state st.
For each discrete Markov state i, the state dependent risk price vector νi is equal to:
νi = γcσc,i (47)
For each discrete Markov state i, the state dependent risk free rate ri is equal to:
ri = −δc1− γc1− ρc
[(ρc − γc1− γc
)hρc−1i − 1
]+ γcµc,i −
1
2γc(1 + γc)|σc,i|2 −
Ns∑j=1
Λijeυ(i,j) (48)
Finally, the SDF relative jump sizes encoded via υ(i, j) are equal to:
υ(i, j) = (ρc − γc) ln
(hjhi
)(49)
Compensations per unit of Brownian risk can be read via the coordinates of the vector νst ,
while jump compensation is encoded via eυ(st−,st)−1. Indeed, take any asset whose valuation
Vt follows:
dVtVt
= µv,tdt+ σv,t · dBt +∑st 6=st−
(eξv(st−,st) − 1
) (dN
(st−,st)t − Λst−,stdt
)Absence of arbitrage imposes that MtVt be a local martingale, in other words E [d (MtVt)] = 0.
Using Ito’s lemma leads to the following pricing restriction:
µv,t − rst = σv,t · νst −∑st 6=st−
(eξv(st−,st) − 1
) (eυ(st−,st) − 1
)Λst−,st
For example, a claim to the creditor’s aggregate consumption Cc,t = Yc,t has a price Pc,t =
Pc,st(Yt) that solve the following HJB equation:
riPc,i(Y ) = Y + (µc,i − νi · σc,i)Y P ′c,i(Y ) +1
2|σc,i|2Y 2P ′′c,i(Y ) +
Ns∑j=1
ΛijPc,j(Y )
The solution to the system above is of the form Pc,i(Y ) = Pc,iY , for a set of constants
Pc,ii≤Ns that are equal to:
Pc =[diagi (ri + νi · σc,i − µc,i)− Λ
]−1
1 (50)
69
Expected excess returns on this claim thus take the following form:
E[d (Pc,stYt) + Ytdt
Pc,stYt− rstdt|Ft
]= νst · σc,stdt−
∑s′
Λst−s′
(Pc,s′
Pc,st−− 1
)(eυ(st−,s′) − 1
)dt
A.1.5 A General Equilibrium Interpretation
In this section, I describe how to obtain, in a general equilibrium setting, the stochastic
discount factor Mt, whose dynamics are given by equation (10). For this, I introduce the
subscript “n” for a country’s identity, and assume a continuum of countries of measure 1. Y nt
is re-interpreted as tax collections of country n – imagine for example that country n has a
flow output Xnt and a tax rate εn, such that Y n
t = εnXnt . Suppose country n’s output evolves
as follows:
dXnt
Xnt
= µstdt+ σst · dBt + σndBnt
Bnt is a country-specific Brownian shock that is independent of Bt, the vector of aggregate
Brownian shocks. Note that all countries need to have the same expected growth rate µst in
all states st, but not necessarily the same output volatility24. The dynamics for aggregate
world output is thus:
dXt =
∫dXn
t dn
=
∫[µstX
nt dt+Xn
t (σst · dBt + σndBnt )] dn
= Xt (µstdt+ σst · dBt)
The latter equality uses the law of large numbers and leverages the fact that Xnt and dBn
t are
not correlated. Given that tax revenues of country n are equal to Y nt = εnXn
t , the government
revenue process Y nt follows the same stochastic differential equation as Xn
t . Government n
resource constraint is the following:
Cnt = Y n
t + Int Dnt − (κ+m)F n
t
Cnt represents government spending, Int represents government debt issuances (debt is issued
at priceDnt ), and F n
t represents the face value of government debt. We assume for the moment
that εn is fixed and exogenous, and that the government will only choose its financing and
24Forcing all countries to have the same expected GDP growth rate also guarantees that countries all“survive” as t→ +∞, in other words no country becomes arbitrarily small asymptotically.
70
default policies. Due to institutional frictions, government n’s utility function is not exactly
equal to the utility function of its citizens; more specifically, the government maximizes the
following objective function (subject to government resource constraint above):
Jnt = E[∫ +∞
t
ϕ (Cns , J
ns ) ds|Ft
]In other words, the government maximizes flow utility over government spending only; more-
over, the government will be more impatient than its citizens: δ, the rate of time preference
of the government, will verify δ > δc, where δc will be the rate of time preference of the
citizens/creditors (to be discussed shortly). Government of country n may elect to default on
its debt. In such case, the government is shut down from capital markets for some exponen-
tially distributed time (parametrized by λ), and suffers a temporary drop in tax collection
efficiency: while in autarky, the government only collects Y nt = αεnXn
t from its citizens, with
α < 1. Upon exit from financial autarky, the government from country n emerges with a
lower debt burden, and recovers its pre-default tax efficiency Y nt = εnXn
t . Note that the drop
in tax collections suffered by the defaulting government is only temporary, and lasts the time
of the capital market’s exclusion.
Government debt is bought by a “representative world investor” – in other words, a citizen
whose equilibrium consumption process is proportional to world consumption. Let Dt be the
set of indices of countries in default at time t, and Dct = [0, 1]\Dt its complement. Remember
that world citizens enjoy flow consumption Γt equal to:
Γt :=
∫n∈Dct
[(1− εn)Xnt + (κ+m)F n
t − Int Dnt ] dn+
∫n∈Dt
(1− αεn)Xnt dn = Xt − Ct
In the above, Ct represents aggregate government spending. Consumption by the “represen-
tative world investor” is simply equal to world output minus taxes paid plus income received
on its government debt portfolio minus investments in government debt. Assume this “rep-
resentative world investor” faces dynamically complete markets, and has preferences over
consumption Γt and government spending Ct as follows:
Ut = E[∫ +∞
t
Ψ (Γs + Cs, Us) ds|Ft]
In other words, consumption and government spending are perfect substitutes in the rep-
resentative investor’s preferences. Given those assumptions, the investor’s equilibrium con-
sumption (via market clearing) is equal to Γt + Ct = Xt, and the investor’s marginal utility
71
is given by equation (45), meaning that the pricing kernel Mt evolves as follows:
dMt
Mt
= −rstdt− νst · dBt +∑st 6=st−
(eυ(st−,st) − 1
) (dN
(st−,st)t − Λst−,stdt
)rs, νs and the SDF jumps υ(s, s′) correspond to those described in Section A.1.4.
A.1.6 Controlled Stochastic Processes Y(τ )t and F
(I,τ )t
An admissible issuance policy Itt≥0 of the government is a progressively measurable process
that is a function of the payoff-relevant variables. In other words, It = Ist(Yt, Ft), for a set
of measurable functions I := Iii≤Ns . I will require the function It to satisfy the standard
integrability condition, for all t ≥ 0, almost surely:
E[∫ t
0
|Is|ds]< +∞
An admissible default policy is an increasing sequence of stopping times (with respect to Ft)noted τ = τkk≥1, which can be written (for k ≥ 0), τk+1 = inft ≥ τk+τe,k : (Yt, Ft) ∈ Ost,for a finite number of Borel sets Oii≤Ns , where τe,kk≥1 is a sequence of i.i.d. exponentially
distributed times (with parameter λ), and where I have set τ0 = τe,0 = 0.
For a given admissible default policy τ ∈ T , define N(τ )d,t := maxk ∈ N : τk ≤ t (resp.
N(τ )e,t := maxk ∈ N : τk + τe,k ≤ t) to be the counting process for default events (resp.
capital markets re-entry events). Let 1(τ )d,t be the default indicator, equal to 1 when the
sovereign government is in default, and zero otherwise:
1(τ )d,t := 1t∈[τNd,t ,τNd,t+τe,Nd,t)
Using this notation, the dynamic evolution of the controlled stochastic process Y(τ )t can be
expressed as follows:
dY(τ )t = Y
(τ )t−
(µstdt+ σst · dBt + (α− 1)dN
(τ )d,t
)(51)
Similarly, the dynamic evolution of the controlled stochastic process F (I,τ ) can be expressed
72
as follows:
dF(I,τ )t =
(It− −mF (I,τ )
t−
)(1− 1
(τ )d,t
)dt
+
(θα exp
[∫ t
τNd,t
(µsu −
1
2|σsu |2
)du+ σsu · dBu
]− 1
)F
(I,τ )t− dN
(τ )e,t (52)
The drift term in the dynamic evolution of F(I,τ )t relates to issuances and debt redemptions
when the government is performing under its debt obligations, while the jump term relates to
reductions in the debt face value in connection with a restructuring and re-entry into capital
markets post-default.
Given a debt price schedule D := Di(·, ·)i≤Ns that the government faces, and given
admissible issuance and default policies (I, τ ) used by the government (where (I, τ ) might
not necessarily be consistent with the debt prices D), there is a controlled flow consumption
process C(I,τ ;D)t :
C(I,τ ;D)t =
[Y
(τ )t + ItDst
(Y
(τ )t , F
(I,τ )t
)− (κ+m)F
(I,τ )t
] (1− 1
(τ )d,t
)+ Y
(τ )t 1
(τ )d,t
The indicator functions in this expression highlight the fact that the government can smooth
consumption via debt issuances and buy-backs when performing, while it is unable to do so
in default.
A.1.7 Monotonicity of Vi
Take a set of debt price functions Di (·, ·)i≤Ns that are homogeneous of degree zero and
decreasing in F . I focus my attention on two initial levels of output, Y (1) and Y (2) > Y (1),
and show that I must have Vi(Y(1), F ;D) < Vi(Y
(2), F ;D). First, take any arbitrary policy
(I, τ ) ∈ I ×T (not necessarily optimal) followed by the government. Following the issuance
policy I and starting in state (Y (2), F ) yields strictly higher flow payoffs at each time t.
Indeed, Y (τ )t |Y0 = Y (2) is almost surely greater than Y (τ )
t |Y0 = Y (1). In addition,
ItDst
(Y
(τ )t , F
(I,τ )t
)|Y0 = Y (2) is almost surely greater than ItDst
(Y
(τ )t , F
(I,τ )t
)|Y0 =
Y (1), since the debt price conditioned on starting in state (Y (2), F ) is almost surely greater
than the debt price conditioned on starting in state (Y (1), F ) (since I assumed that the debt
price is decreasing in the debt face value). Thus, C(I,τ ;D)t |Y0 = Y (2) is almost surely
greater than C(I,τ ;D)t |Y0 = Y (1), which means that the life-time utility is increasing in
Y , for any arbitrary issuance and default policy. Thus, the supremum over all feasible
issuance and default policies, Vi (·, ·;D), is also increasing in output Y . The proof for the
monotonicity of Vi (·, ·;D) in F is identical, since consumption C(I,τ ;D)t is decreasing in the
73
level of indebtedness F and since Di(·, ·) is decreasing in F .
I then show that the optimal issuance policy is homogeneous of degree 1 and the optimal
default policy is barrier type. Take an arbitrary (Y, F ), and the related optimal Markov
issuance and default policies (I∗Y,F , τ∗Y,F ) = arg max Ji (Y, F ; (I, τ );D). Take ε > 0, and
focus on starting output and face value levels (εY, εF ). Consider the policy (IεY,εF , τεY,εF ),
such that IεY,εF = εI∗Y,F , and τεY,εF = τ ∗Y,F . Since (I∗Y,F , τ
∗Y,F ) is feasible conditioning on
(Y0, F0) = (Y, F ), since the output dynamics are linear in Y and since the debt face value
dynamics are homogeneous of degree 1 in (Y, F ), it must be the case that (IεY,εF , τεY,εF ) is
feasible conditioning on (Y0, F0) = (εY, εF ). Thus, I have:
Ji(εY, εF ; (IεY,εF , τεY,εF );D) ≤ Vi(εY, εF ;D)
Then assume for a second that the inequality above is strict. If that was the case, then take
(I∗εY,εF , τ∗εY,εF ) = arg max Ji (εY, εF ; (I, τ );D). Consider the policy (IY,F , τY,F ), such that
IY,F = I∗εY,εF /ε, and τY,F = τ ∗εY,εF . Then is it immediate to see that this policy is feasible
conditioned on starting at (Y0, F0), and it is also immediate to see that:
Ji(Y, F ; (IY,F , τY,F );D) > Ji(Y, F ; (I∗Y,F , τ∗Y,F );D) = Vi(Y, F ;D)
This is a contradiction. Thus, the optimal issuance policy is homogeneous of degree 1 in
(Y, F ). Since the value function is decreasing in F and increasing in Y , the default policy
must be a barrier default policy.
A.1.8 Monotonicity of Di
Let (I, τ ) ∈ I × T be admissible issuance and default Markov policies. Assume that I is
homogeneous of degree 1 in (Y, F ) and that τ is barrier. Note Ii(Y, F ) = ιi(F/Y )Y , where
ιi(x) := Ii(1, x), for each Markov state i ≤ Ns. Given these assumptions, the default policy
can be written:
τ = inft ≥ 0 : Ft ≥ Ytxst
= inft ≥ 0 : xt ≥ xst
74
Using Ito’s lemma, xt := Ft/Yt follows the following stochastic differential equation under Q:
dx(ι,τ )t =
(1− 1
(τ )d,t
) [(ιst
(x
(ι,τ )t
)−(m+ µst − |σst |2 − νst · σst
)x
(ι,τ )t
)dt− x(ι,τ )
t σst · dBt
]+ x
(ι,τ )t−
(1
α− 1
)dN
(τ )d,t + x
(ι,τ )t− (θα− 1) dN
(τ )e,t
The debt price is an expected present value of flow payoffs. Since such flow payoffs are
homogeneous of degree zero in (Y, F ), and since the default policy is barrier, the debt
price function must be homogeneous of degree zero. With an abuse of notation, I will note
Di (Y, F ; (ι, τ )) = Di (x; (ι, τ )). Note then that the debt price cannot be greater than the
price Drfi of a risk-free claim to sovereign debt cash-flows, where Drf verifies:
Drf = (κ+m)(
diagi (ri +m)− Λ)−1
1 (53)
I then use a result that will be proven in Section A.1.9: the fact that the defaulted debt price
must satisfy:
Dd(x) = λθαΞ−1D(θx)
The matrix Ns × Ns matrix Ξ is equal to diagi (ri + νi · σi + λ− µi) − Λ. I then introduce
the operator T, defined for any Ns × 1 vector f of continuous decreasing functions whose
ith coordinate is fi : (0,maxj xj) → [0, Drfi ] (Drf
i is the price of risk-free debt is state i, see
equation (53)) as follows. If x ≥ xi, set (Tf)i (x) = λθα [Ξ−1f(θx)]i. If x ≤ xi, then:
(Tf)i (x) : = Ei,x[∫ τ
0
e−∫ t0 (rsu+m)du(κ+m)dt+ λθαe−
∫ τ0 (rsu+m)du
[Ξ−1f(θxτ )
]sτ
](54)
= Drfi + Ei,x
[e−
∫ τ0 (rsu+m)du
(λθα
[Ξ−1f(θxτ )
]sτ−Drf
sτ
)](55)
Given assumption 2, given that θ < 1 and α < 1, and given that the function fi has an image
in [0, Drfi ], it must be the case that λθα [Ξ−1f(·)]i ≤ Drf
i for any state i, which means that
the term in brackets in equation (55) is negative. Thus, (Tf)i (·) is a decreasing function
of x, positive and bounded above by Drfi . The Feynman-Kac theorem also provides for the
continuity of the function (Tf)i (·). Thus, T maps Ns × 1 vectors of continuous bounded
decreasing functions with image in [0, Drfi ] into itself. For any pair of vectors of functions
75
f1,f2 whose components are continuous and decreasing on the interval [0,maxi xi], I have:
| (Tf2 − Tf1)i |(x) = λθαEi,x[e−
∫ τ0 (rsu+m)du
∣∣ [Ξ−1 (f2(θxτ )− f1(θxτ ))]sτ
∣∣]≤ λθα||Ξ−1|| · |f2 − f1|∞
Since λθα||Ξ−1|| < 1 (given assumption 2), T is a contraction, and the contraction map-
ping theorem provides for a unique continuous, bounded and decreasing vector of functions
vector with ith row vi(x), and if I note vd(x) the Ns × 1 vector with ith row vdi (x), I need to
solve the non-linear equation:
Υvd(x)− λv(θx) =δ
1− ρ[(1− γ)vd(x)
] ρ−γ1−γ (56)
In the above[(1− γ)vd(x)
] ρ−γ1−γ is to be understood as an element-by-element power function.
Note that vdi (x) admits the integral representation (this will be useful in connection with my
verification theorem):
vdi (x) = Ei[∫ τe
0
e−1−γ1−ρ
∫ t0 Asudu
δ
1− ρ[(1− γ)vst(x)]
ρ−γ1−γ dt+ e−
1−γ1−ρ
∫ τe0 Asuduvsτe (θx)
](57)
79
Note that for γ = ρ, I can solve equation (56) explicitly, obtaining:
vd(x) = Υ−1
(λv(θx) +
δ
1− γ1
)In the above, 1 is a Ns × 1 vector of ones. Default optimality can be written Vi(Y, F ) ≥V di (Y, F ), which, stated in terms of the normalized value functions, can be written:
vst(xt)− α1−γvdst(xt) ≥ 0 ∀t ≥ 0
Applying Ito’s lemma to vst(xt)− α1−γvdst(xt), the diffusion term is equal to:
−xt(v′st(xt)− α
1−γ (vdst)′ (xt))σst · dBt
In particular, since vst(xst) = α1−γvdst(xst), the only way for the inequality vst(xt)−α1−γvdst(xt) ≥0 to be preserved in the presence of Brownian shocks at the default boundary is for the dif-
fusion term above to be identically zero at such boundary. This leads to the smooth-pasting
optimality condition:
v′i(xi) = α1−γ (vdi )′ (xi)Differentiating the implicit equation defining vd(x) w.r.t. x, I obtain the following expression
for(vd)′
(xi):
(vd)′
(xi) = λθ
(Υ + δ
γ − ρ1− ρ
diagj
([(1− γ)vdj (xi)
]− 1−ρ1−γ))−1
v′(θxi)
A.1.12 Verification Theorem
Let vii≤Ns be a family of functions such that for each i, vi ∈ C1(R+) ∩ C2(R+ \ xi)satisfies the assumptions of the theorem. Let (ι, τ ) ∈ I × T be an arbitrary policy, I have
80
the following Ito formula:
e−1−γ1−ρ
∫ t0 Asuduvst(xt) = vi(x)−
∫ t
0
e−1−γ1−ρ
∫ z0 Asudu
(1− 1
(τ )d,z
)xzv
′sz(xz)σsz · dBz
+
∫ t
0
e−1−γ1−ρ
∫ z0 Asudu
∑s′ 6=sz−
(vs′(xz)− vsz−(xz)
) (dN sz−,s′
z − Λsz−,s′dz)
+
∫ t
0
e−1−γ1−ρ
∫ z0 Asudu
(1− 1(τ )d,z
)L(ι)vsz(xz) +
∑s′ 6=sz−
Λsz−,s′(vs′(xz)− vsz−(xz)
)− 1− γ
1− ρAszvsz(xz)
dz+
∫ t
0
e−1−γ1−ρ
∫ z0 Asudu
(vsz(xz)− vsz−(xz−)
)dN
(τ )d,z +
∫ t
0
e−1−γ1−ρ
∫ z0 Asudu
(vsz(xz)− vsz−(xz−)
)dN (τ )
e,z
See for example Protter (2005). For the arbitrary control policy (ι, τ ), I note c(ι,τ )st (xt)
the resulting consumption-to-output policy. Using Ito’s lemma above, using the variational
inequality in the assumption of the theorem, and using equation (57):
e−1−γ1−ρ
∫ t0 Asuduvst(xt) ≤ vi(x)−
∫ t
0
e−1−γ1−ρ
∫ z0 Aszdzδ
c(ι,τ )sz (xz)
1−ρ [(1− γ)vsz(xz)]ρ−γ1−γ
1− ρdz
+
∫ t
0
e−1−γ1−ρ
∫ z0 Asudu
∑s′ 6=sz−
(vs′(xz)− vsz−(xz)
) (dN sz−,s′
z − Λsz−,s′dz)
−∫ t
0
e−1−γ1−ρ
∫ t0 Aszdz
(1− 1
(τ )d,z
)xzv
′sz(xz)σsz · dBz
The terms on the second and third line above are martingales since vi and v′i are bounded.
Thus, taking expectations on both sides of this equality, I obtain:
Ei,x[∫ t
0
e−1−γ1−ρ
∫ z0 Aszdzδ
c(ι,τ )sz (xz)
1−ρ [(1− γ)vsz(xz)]ρ−γ1−γ
1− ρdz
]+Ei,x
[e−
1−γ1−ρ
∫ t0 Aszdzvst(xt)
]≤ vi(x)
Taking t → +∞, using the assumption that limt→+∞ inf e−∫ t0
1−γ1−ρAsuduvst
(x
(ι,τ )t ;D
)≤ 0,
and using the monotone convergence theorem, I then obtain the desired result: vi(x;D) ≥Ji (1, x; (ι, τ );D) for any admissible control policy. The proof of the second part of the
theorem relies on steps identical to those described above, except that inequalities are now
replaced by equalities. The uniqueness of vi(·;D) as a solution to re-scaled sequence problem
equation (16), shows that vi(x;D) = Ji (1, x; (ι∗, τ ∗) ;D).
81
A.1.13 Sketch of Equilibrium Existence Proof
I discuss here a possible route to prove that an equilibrium exists in a simpler environment
without discrete SDF states, and where the punishment upon default is financial autarky for-
ever. Upon a sovereign default, creditors’ recovery value is zero. In this simpler environment,
the Markov perfect equilibrium features only the debt-to-GDP ratio as a state variable. Take
an arbitrary debt price schedule D : R+ → [0, κ+mr+m
] that is continuous and strictly decreasing
on that interval. Given this debt price schedule, construct the sovereign’s “best response”,
in other words construct the value function v(·;D) as well as the optimal issuance and de-
fault policies ι∗(·;D) and x∗(D). This best response exists: given a debt price schedule D,
the function v(·;D) is simply the optimal life-time utility in a single-agent optimal control,
optimal stopping problem, where the control is the issuance rate ι and the stopping time is
the default time τ . Using those issuance and default policies, construct a new debt price
schedule D (·; (ι∗(·;D), x∗(D))).
I have implicitly constructed a functional map T, which takes a continuous decreasing
functionD : R+ → [0, κ+mr+m
] and maps it into a continuous decreasing functionD (·; (ι∗(·;D), x∗(D))) :
R+ → [0, κ+mr+m
]. In fact, by studying the sovereign’s behavior when the debt price is constant
and equal to its risk-free value κ+mr+m
, I can restrict this functional map to functions defined
on the interval [0, xrf ] (where xrf is the optimal sovereign default boundary when the debt
is priced at its risk-free value by creditors). Indeed, it is straightforward to show that the
default boundary must be decreasing in the debt price schedule – in other words, if for any
x, D1(x) ≥ D2(x), it must be the case that x∗(D1) ≥ x∗(D2). A Markov perfect equilibrium
of my economy is simply a fixed point of the functional map constructed. Schauder’s fixed
point theorem (appropriate for infinite dimensional spaces) could then be invoked in order
to establish the existence of a fixed point of such functional map. An appropriate space of
functions to use is any subset that is closed, bounded and equicontinuous. Indeed, since
[0, xrf ] is compact, Arzela-Ascoli’s theorem guarantees that any such subspace of functions
is compact. A good candidate to restrict oneself would be the space of Lipschitz continuous
functions that have the same Lipschitz constant. In order to apply Schauder’s fixed point
theorem, two theoretical hurdles thus have to be overcome. First, one would need to show
that the mapping T is continuous. Second, one would need to show that the mapping T pre-
serves Lipschitz continuity. Once those two conditions are established, existence of a Markov
perfect equilibrium is straightforward.
82
A.1.14 Expected Default Times and Ergodic Distribution
I note Ti(x), the risk-natural expected default time conditioned on the debt-to-GDP ratio
being equal to x and the state st = i. Mathematically, the expected default time can be
written Ti(x) := Ei,x [τ ]. Using Feynman-Kac, it is immediate to show that Ti(·) solves the
following HJB equation, for x ∈ (0, xi):
0 = 1 +(ιi(x)−
(m+ µi − |σi|2
)x)T ′i (x) +
1
2|σi|2x2T ′′i (x) +
Ns∑j=1
ΛijTj(x) (58)
The 2×Ns boundary conditions consist in (i) value mathing conditions at the default bound-
ary and (ii) Robin boundary conditions at x = 0:
0 = Ti(x) ∀x ≥ xi
0 = 1 + ιi(0)T ′i (0) +Ns∑j=1
ΛijTj(0)
I then focus on the ergodic measure fi of the state variable under the physical measure P,
conditioned on being in state i and conditioned on the government being performing under
its debt obligations (I emphasize the word measure as opposed to density since fi does not
integrate to 1). For x ∈ (0, xi) and x /∈ θxj1≤j≤Ns , fi solves the following Kolmogorov-
forward equation:
0 = − d
dx
[(ιi(x)−
(m+ µi − |σi|2
)x)fi(x)
]+
1
2
d2
dx2
[|σi|2x2fi(x)
]+
Ns∑j=1
Λjifj(x)
The equation above is not applicable at the points θxj1≤j≤Ns (the points of re-entry of the
sovereign following a “smooth” default, i.e. a default such that xτ− = xsτ ), but the measures
fi are continuous at those points. The following boundary condition holds at x = xi:
fi(xi) = 0 (59)
This equation can be obtained heuristically by approximating the continuous time process
xt by a discrete state Markov chain, and analyzing transitions in and out of the state
x = xst between time t and time t + ∆t. It is also a standard condition for absorbing
boundaries. I note gi the fraction of time the sovereign is in autarky in SDF state i – note
that such fraction does not depend on the debt-to-GDP ratio at entry into the default state
given the memory-less property of the stochastic process st, and given that the autarky time
83
length is exponentially distributed and independent of the process st. In other words, if
g :=∑Ns
i=1 gi, then I have:
gig
=Ei[∫ τe
01st=idt
]Ei [τe]
=
((I − 1
λΛ
)−1)ii
(60)
For any Markov state s = i, the integral of the ergodic distribution over the state space [0, xi),
in addition to the expected time spent in autarky gi, need to add up to πi, the stationary
measure of the process st: ∫ xi
0
fi(x)dx+ gi = πi (61)
Thus, equations (59) and (61) gives me 2×Ns “boundary” conditions, allowing me to solve for
the Ns Kolmogorov-forward equations, which are second order ordinary differential equations.
However, the constants gi1≤i≤Ns in equation (60) are only determined up to the constant g,
which represents the average percentage of time the sovereign spends in autarky post-default.
I determine the constant g numerically via a Markov chain approximation method described
in Section A.3.
Finally, note that in the particular case where there is only one discrete Markov state,
I can derive a pseudo-closed form expression for the stationary measure f . Indeed, in such
case, the ergodic measure f of the state variable under the physical probability measure Psolves the following Kolmogorov-forward equation, valid for x ∈ (0, θx) ∪ (θx, x):
0 = − d
dx
[(ι(x)−
(m+ µ− |σ|2
)x)f(x)
]+
1
2
d2
dx2
[|σ|2x2f(x)
](62)
f is continuous at x = θx (the point of re-entry of the sovereign post-autarky). At x = x,
the ergodic distribution must satisfy the absorbing boundary condition:
f(x) = 0
Equation (62) can be integrated out as follows. For x ∈ (θx, x), I have:
0 = G+ ((m+ µ)x− ι(x)) f(x) +1
2|σ|2x2f ′(x) (63)
The constant G is the “flow” of the density in the positive direction. Using f(x) = 0, I can
integrate equation (63) to obtain, for x ∈ (θx, x):
f(x) =
∫ x
x
exp
[∫ t
x
2
σ2s2((m+ µ)s− ι(s)) ds
]2G
σ2t2dt
84
I also know that the density f is continuous at θx (even though it is not differentiable at that
point). At x = 0, the density must be zero. Indeed, note that in a neighborhood of x = 0,
the stochastic process xt behaves similarly to a geometric Brownian motion to which a
constant strictly positive drift ι(0) has been added, and it is straightforward to show that
the stationary distribution of a geometric Brownian motion which, at x = x, is “reset” to
x = θx, admits a stationary density with value zero at x = 0. Thus, on [0, θx), the density
f takes the following form:
f(x) = exp
[∫ θx
x
2
σ2s2((m+ µ)s− ι(s)) ds
]f(θx)
This integration provides for the continuity of f at x = θx. Finally, the integral of the ergodic
measure over the state space [0, x], in addition to the expected percentage of time spent in
autarky, need to add up to 1: ∫ x
0
f(x)dx+1/λ
1/λ+ T (θx)= 1
This pins down the unknown constant G.
A.1.15 Credit Spreads
I leverage the equation that the credit spread ςi(x) satisfies:
Di(x) = (m+ κ)
[(diagj (rj + ςi(x) +m)− Λ
)−1
1
]i
Some algebra can show that for any state i, I have:
ς ′i(x) = − D′i(x)
(κ+m)
[(diagj (rj + ςi(x) +m)− Λ
)−2
1
]i
Since the debt price function Di is decreasing in the debt-to-GDP ratio x and since the
denominator in the expression above is positive, ς ′i < 0. Some algebra also shows that the
function xς ′i(x) can be expressed as follows:
d
dx(xς ′i(x)) = ς ′i(x) +
(κ+m)x (ς ′i(x))2
[(diagj (rj + ςi(x) +m)− Λ
)−3
1
]i
− xD′′i (x)
(κ+m)
[(diagj (rj + ςi(x) +m)− Λ
)−2
1
]i
85
I have showed previously that ς ′i > 0. The second term is positive, and the third term is also
positive if the debt price function Di is concave.
A.1.16 Credit Default Swap Premia
As specified in the main text, I define the risk-neutral present value of future credit losses
and the risk-neutral present value of future CDS premia as follows:
Li(x, T ) : = Ex,i[1τ<Te
−∫ τ0 rsudu max (0, 1−Dsτ (xτ ))
]Pi(x, T ) : = Ex,i
[∫ T∧τ
0
e−∫ t0 rsududt
]The CDS premium is simply the ratio of those two quantities: ςi(x, T ) = Li(x,T )
Pi(x,T ). An applica-
tion of Feynman-Kac leads to the following partial differential equations satisfied by Li and
Pi, for 1 ≤ i ≤ Ns:
riLi(x, t) = −∂Li∂t
(x, t) + LiLi(x, t) +Ns∑j=1
ΛijLj(x, t)
riPi(x, t) = 1− ∂Pi∂t
(x, t) + LiPi(x, t) +Ns∑j=1
ΛijPj(x, t)
The boundary conditions are as follows, for t ∈ [0, T ):
Li(x, 0) = 0 ∀x < xi
Pi(x, 0) = 0 ∀x < xi
Li(x, t) = 1−Ddi (x) ∀x ≥ xi
Pi(x, t) = 0 ∀x ≥ xi
I can then compute the expected excess return and the return volatility on a T -maturity CDS
contract. Imagine that at time t, an investor sells protection on the specific sovereign credit,
for $1 dollar notional amount and using a T -maturity contract. At time t, no cash-flow is
exchanged, the value of the CDS contract is zero and the premium agreed upon between
the buyer and the seller is equal to ςst(xt, T ) = Lst(xt, T )/Pst(xt, T ). At time t + dt, the
protection seller has accrued ςst(xt, T )dt of premium income. The value of the “premium
leg” of his CDS contract is now equal to ςst(xt, T )Pst+dt(xt+dt, T − dt) while the value of the
“default leg” of his CDS contract is now equal to Lst+dt(xt+dt, T − dt). In other words, his
86
excess return (computed based on a $1 notional risky investment) is equal to:
dRet,T = ςst(xt, T )dt+ ςst(xt, T )Pst+dt(xt+dt, T − dt)− Lst+dt(xt+dt, T − dt)
This return is viewed as an excess return since the protection seller did not put any money
upfront to enter into his contract. To compute CDS expected excess returns and return
volatilities, I use Ito’s lemma and the relationship dRet,T = ςtdt + ςtPst+dt (xt+dt, T − dt) −
Lst+dt (xt+dt, T − dt):
dRet,T = ςtdt+ ςt
(Pst + LstPstdt−
∂Pst∂t
dt− xt∂Pst∂x
σst · dBt +∑s′
(Ps′ − Pst) dN(st,s′)t
)
−
(Lst + LstLst −
∂Lst∂t
dt− xt∂Lst∂x
σst · dBt +∑s′
(Ls′ − Lst) dN(st,s′)t
)
I then use the relationship between the operators Lst and Lst :
Lst = Lst + xtνst · σst∂
∂x
Using ςt = Lst/Pst and the HJB equation satisfied by Lst and Pst , I have:
dRet,T = −Lst
∑s′
Λst−,s′
(eυ(st−,s′) − 1
)( Ps′
Pst−− Ls′
Lst−
)dt−Lst
(xt
∂Pst∂x
Pst−xt
∂Lst∂x
Lst
)νst·σstdt
+ Lst∑s′
(Ps′
Pst−− Ls′
Lst−
)(dN
(st−,s′)t − Λst−,s′dt
)− Lst
(xt
∂Pst∂x
Pst−xt
∂Lst∂x
Lst
)σst · dBt
This leads to the following expression for conditional expected excess returns and conditional
return volatilities:
E[dRe
t,T |Ft]
= −
[(xt
∂Pst∂x
Pst−xt
∂Lst∂x
Lst
)νst · σst +
∑s′
Λsts′
(eυ(st,s′) − 1
)(Ps′Pst− Ls′
Lst
)]Lstdt
var[dRe
t,T |Ft]
=
(xt ∂Pst∂x
Pst−xt
∂Lst∂x
Lst
)2
|σst|2 +∑s′
Λsts′
(Ps′
Pst− Ls′
Lst
)2L2
stdt
87
A.1.17 Consumption Growth vs. Output Growth Volatility
Let me note µxi (xt) the drift rate of xt in SDF regime i, and σxi (xt) its volatility vector:
µxi (xt) := ιi(xt)−(µi +m− γ|σi|2
)xt
σxi (xt) := −xtσi
Using Ito’s lemma, I can compute consumption growth volatility as follows:
dCtCt
=
[c′st(xt)
cst(xt)µxst(xt) +
1
2|σxst(xt)|
2 c′′st(xt)
cst(xt)+ µst +
1
2|σst|2 +
c′st(xt)
cst(xt)σxst(xt) · σst
]dt
+∑s′
(cs′(xt)
cst(xt)− 1
)dN
(st,s′)t +
(c′st(xt)
cst(xt)σxst(xt) + σst
)· dBt
In other words, conditioned on being in SDF regime st, the ratio of consumption growth
volatility to output growth volatility has the following simple expression:
var[dCtCt
∣∣Ft]var[dYtYt
∣∣Ft] =
(1−
xtc′st(xt)
cst(xt)
)2
+1
|σst|2∑s′
Λst,s′
(cs′(xt)
cst(xt)− 1
)2
Thus, the ratio of consumption growth volatility to output growth volatility crucially depends
on the elasticity of the consumption function w.r.t. the debt-to-GDP ratio. Moreover,
since the consumption function cst(·) is decreasing in the debt-to-GDP ratio, it turns out
that consumption growth volatility is greater than output growth volatility. Consumption
volatility is also enhanced by the SDF shocks. It is also immediate to verify that the presence
of SDF shocks breaks the unit correlation between consumption growth and output growth.
Such correlation is equal to:
corr
[dCtCt
,dYtYt
∣∣Ft] =1√
1 +∑
s′ Λst,s′
(cst (xt)−cs′ (xt)cst (xt)−xtc′st (xt)
)2< 1
A.1.18 Risk-Neutral Government
Assume γ = ρ = 0, and assume that µi = µ and σi = σ for all i ≤ Ns. The government is
risk-neutral; its incentive to take on debt is solely due to the fact that it is more impatient
that its creditors: δ > ri, for all state i ≤ Ns. In any equilibrium with default where the
88
issuance policy ιi is finite, the HJB equation for the government life-time utility takes the
following form:
(δ − µ) vi(x)−Ns∑j=1
Λijvj(x) = maxιi
[δ (1 + ιiDi(x)− (κ+m)x) + [ιi − (µ+m)x] v′i(x) +
1
2|σ|2x2v′′i (x)
]
This expression is linear in ιi, meaning that for an equilibrium to exist with a finite smooth
issuance policy, it must be the case that δDi(x) + v′i(x) = 0. Reinjecting this condition into
the HJB above leads to:
(δ − µ) vi(x)−Ns∑j=1
Λijvj(x) = δ (1− (κ+m)x)− (µ+m)xv′i(x) +1
2|σ|2x2v′′i (x)
In this HJB equation, the issuance policy and the debt price have disappeared, and neither
constants nor boundary conditions are dependent on the SDF state. Thus, one solution to the
HJB equation is to have vi(x) = v(x) for all state i ≤ Ns. Of course in that case, the default
boundaries are also SDF state-independent, in other words for all i ≤ Ns, xi = x. Finally,
since δDi(x) + v′i(x) = 0, it is also the case that the debt price is SDF state-independent. v
then solves the following:
(δ − µ) v(x) = δ (1− (κ+m)x)− (µ+m)xv′(x) +1
2|σ|2x2v′′(x) (64)
In other words, the govermnent life-time utility is identical to its value if it was allowing its
debt to amortize, without ever re-issuing new debt or buying back existing debt. The second
order ordinary differential equation admits the following characteristic polynomial:
1
2|σ|2ξ2 −
(m+ µ+
1
2|σ|2
)ξ − (δ − µ) = 0
Let ξ be the positive root of such polynomial (the other root being strictly negative):
ξ :=1
2
(1 +
2(m+ µ)
|σ|2
)[1 +
(1 +
8(δ − µ)|σ|2
(2(m+ µ) + |σ|2)2
)1/2]> 1
Since the value function must be finite at x = 0, it takes the following form:
v(x) =δ
δ − µ− δ
(κ+m
δ +m
)x+ kv
(xx
)ξ
89
kv is a constant of integration that will be found using boundary conditions. At default,
v(x) = αvd(x), where the constant vd(x) satisfies:
vd(x) =δ + λv(θx)
δ + λ− µ
Thus the constant kv solves:
δ
δ − µ− δ
(κ+m
δ +m
)x+ kv =
αδ
δ + λ− µ+
αλ
δ + λ− µ
[δ
δ − µ− δ
(κ+m
δ +m
)θx+ kvθ
ξ
]The optimal default boundary x then satisfies the smooth pasting condition:
v′(x) = α(vd)′(x) =αθλv′(θx)
δ + λ− µ
This leads to a value of x that verifies:
δ
(κ+m
δ +mx
)(1− αθλ
δ + λ− µ
)= kvξ
(1− αθξλ
δ + λ− µ
)I then deduce the following optimal default boundary x and the constant of integration kv:
x =ξ
ξ − 1
(δ +m
κ+m
)( 1−αδ−µ
1− αθλδ+λ−µ
)
kv =
(δ
ξ − 1
)( 1−αδ−µ
1− αθξλδ+λ−µ
)
The debt price is computed using the equality δD(x) + v′(x) = 0:
D(x) =κ+m
δ +m− ξkv
x∗
(xx
)ξ−1
=
(κ+m
δ +m
)[1−
(1− αθλ
δ+λ−µ
1− αθξλδ+λ−µ
)(xx
)ξ−1]
D is of course a decreasing function of x since ξ > 1. This is the condition that DeMarzo
and He (2014) uncover as the necessary and sufficient condition for optimality of a “smooth”
financing strategy for the government. It is then easy to show that:
v(x) = δ
[1
δ − µ
(1−
(1− α
1− αθξλδ+λ−µ
)(xx
)ξ)− xD(x)
]
90
This formula has a natural interpretation: the life-time utility for the government is equal
to the present value (from the government’s perspective) of its endowment stream, adjusted
for expected welfare losses due to default, minus the aggregate value of sovereign debt. The
debt price function D must satisfy the HJB equation (from Feynman-Kac):
(ri +m)D(x) = κ+m+[ιi(x)−
(µ+m− |σ|2 − σ · νi
)x]D′(x) +
1
2|σ|2D′′(x) (65)
Take equation (64), differentiate w.r.t. x, and use δD(x) + v′(x) = 0 to obtain:
(δ +m)D(x) = κ+m−(µ+m− |σ|2
)xD′(x) +
1
2|σ|2D′′(x) (66)
I can interpret probabilistically this HJB equation: the price of one unit of debt is equal to
the expected discounted net present value of interest and principal repayments on such debt
contract, where the discount rate is δ, and where the default time is the first time at which
the debt-to-GDP ratio hits the boundary x, using a probability measure under which no new
debt is ever issued by the government. Subtract Equations (66) from (65), and simplify to
obtain:
ιi(x) =(δ − ri)D(x) + σ · νixD′(x)
−D′(x)
=δ − riξ − 1
[(1− αθξλ
δ+λ−µ
1− αθλδ+λ−µ
)( xx
)ξ−1
− 1
]x− σ · νix
The time-varying interest rates and prices of risk only impact the financing policy of the
government: in periods of high risk-prices or relatively high risk-free rates, the government
adjusts its financing policy downwards. Finally, note that the stochastic differential equation
for xt takes the following form (in the continuation region):
dxt =
(δ − riξ − 1
(1− αθξλ
δ+λ−µ
1− αθλδ+λ−µ
)xξ−1x2−ξ
t −(δ − riξ − 1
+m+ µ− |σ|2 + σ · νi)xt
)dt− xtσ · dBt
This equation admits a singularity at x = 0, since at that point, the drift rate is unbounded,
except when ξ ∈ (1, 2].
91
A.1.19 Endogeneous Growth
The dynamic equations for the state variables and the resource constraint are as follows:
dKt = (Ht − ηKt)dt+Ktσ · dBt
dFt = (It −mFt)dt
Ct +Ht = aKt + ItDt − (κ+m)Ft −G(Ht, Kt)
By noting ιt := It/Kt, ht := Ht/Kt, a natural state variable arises: xt := Ft/Kt, which
evolves according to:
dxt =(ιt −
(m+ ht − η − |σ|2
)xt)dt− xtσ · dBt
As usual, the value function, in SDF state i, can be written Vi(K,F ) = vi (x)K1−γ. The
HJB equation satisfied by vi in the continuation region can be written:
1− γ1− ρ
(δ + (1− ρ)
(η +
1
2γ|σ|2
))vi(x)−
Ns∑j=1
vj(x) =
maxιi,hi
[δ
1− ρ(a+ ιiDi(x)− (κ+m)x− (hi + g(hi)))
1−ρ [(1− γ)vi(x)]ρ−γ1−γ
+hi(1− γ)vi(x) +[ιi −
(m+ hi − η − γ|σ|2
)x]v′i(x) +
1
2|σ|2x2v′′i (x)
]The (necessary and sufficient) first order conditions for investment and debt issuances are as
For xk ≥ xs, the sovereign government is in default and I compute D(i,j)s (xk) as follows:
D(i,j)s (xk) = λθα
(Ξ−1D(i,j)(θxk)
)s
This is a linear system ofNs×(Nh+1) equations inNs×(Nh+1) unknown D(i,j)s (xk)0≤k≤Nh,s≤Ns
which can be solved easily via a simple matrix inversion. Note that the matrix to be inverted
100
is sparse, which greatly reduces computing time.
I then describe how to compute v(i,j)s in each discrete Markov state s, given an issuance
schedule ι(i,j), a debt price schedule D(i,j) and a default policy x(i). Once again I omit the
superscript (i, j) when possible. Given a vector of cutoffs x, an issuance policy ι and a debt
price schedule D, the dynamic evolution of the state variables xt and st under the probability
measure induced via Pr(A) = E[e(1−γ)
∫ t0 σsu ·dBu− 1
2(1−γ)2
∫ t0 |σsu |2du1A
]can be expressed as
follows, for xt ∈ [0, xst ]:
dxt =(ιst(xt)−
(m+ µst − γ|σst |2
)xt)dt− xtσst · dBt
= µP(xt, st)dt− xtσst · dBt
dst =∑s′ 6=st−
(s′ − st−)dNst−,s′
t
By introducingQhP(x, s) and ∆thP(x, s), computed in a similar fashion toQh
Q(x, s) and ∆thQ(x, s),
I can construct a new Markov Chain XhP,nn≥0 that approximates the process (xt, st)t≥0
under the probability measure Pr. The transition probabilities of this Markov chain will
satisfy consistency conditions similar to those of equations (69), (70), (71), (72) and (73).
For xs > xk ≥ 0, the sovereign government is performing and I compute v(i,j)s (xk) as follows:
v(i,j)s (xk) =
δ
1− ρ(1 + ιs(xk)Ds(xk)− (κ+m)xk)
[(1− γ)v(i,j)
s (xk)] ρ−γ
1−γ ∆thP(xk, s)
+ e−1−γ1−ρAs∆t
hP
(xk,s)∑x′P,s′
P
Pr(X ′P|xk, s
)v
(i,j)
s′P
(x′P)
For xk ≥ xs, the sovereign government is in default and I compute v(i,j)s (xk) = α1−γ (vds)(i,j)
(xk)
as follows:
(vds)(i,j)
(xk) =
[Υ−1
(δ
1− ρ
[(1− γ)
(vd)(i,j)
(xk)] ρ−γ
1−γ+ λv(i,j)(θxk)
)]s
Note that the resulting system of Ns × (Nh + 1) equations in Ns × (Nh + 1) unknown
v(i,j)s (xk)0≤k≤Nh,s≤Ns is not linear. In order to solve such system, I use a simple proce-
dure: starting with a guess v(i,j,m)(xk)0≤k≤Nh , I iterate, for xk < xs, on the following:
v(i,j,m+1)s (xk) =
δ
1− ρ(1 + ιs(xk)Ds(xk)− (κ+m)xk)
[(1− γ)v(i,j,m)
s (xk)] ρ−γ
1−γ ∆thP(xk, s)
+ e−1−γ1−ρAs∆t
hP
(xk,s)∑x′P,s′
P
Pr(X ′P|xk, s
)v
(i,j,m+1)
s′P
(x′P)
101
For xk ≥ xs, the iteration becomes:
(vds)(i,j,m+1)
(xk) =
[Υ−1
(δ
1− ρ
[(1− γ)
(vd)(i,j,m+1)
(xk)] ρ−γ
1−γ+ λv(i,j,m)(θxk)
)]s
The iterative procedure is stopped once ||v(i,j,m+1) − v(i,j,m)||∞ is sufficiently small. Once
v(i,j) and D(i,j) are computed, I can update the issuance policy as follows, in each state s:
ι(i,j+1)s (xk) = $ι(i,j)s (xk) +
1−$D
(i,j)s (xk)
δD(i,j)
s (xk)[(1− γ)v
(i,j)s (xk)
] ρ−γ1−γ
−(v
(i,j)s
)′(xk)
1/ρ
+ (κ+m)xk − 1
In the above $ ∈ (0, 1) is a dampening parameter that “smoothes” the transition from ι
(i,j)s
to ι(i,j+1)s and prevents infinite loops between debt price and issuance policy26. The derivative(
v(i,j)s
)′(xk) is computed by using a centered finite difference approximation. I iterate on the
inner loop until ||ι(i,j+1) − ι(i,j)||∞ is sufficiently small.
At the conclusion of the inner loop, I have obtained v(i),D(i), ι(i), all assuming a default
policy x(i). I then set x(i+1) by checking the smooth pasting condition at x(i)s for all discrete
states s:
(v(i))′(x(i)s ) ≷ α1−γλθ
[(Υ + δ
γ − ρ1− ρ
diagj
([(1− γ)vdj (xs)
]− 1−ρ1−γ))−1
v′(θxs)
]s
Depending on whether the left handside is greater or less than the right handside, I update
x(i+1)s using a binomial search method.
Once the optimal default boundary x and the optimal issuance policy ι are known, I
can compute the expected default time and the ergodic density of x. The computation of
the expected default time T (x) follows the same logic as the computation of the debt price,
except that the Markov transition probabilities are adjusted to reflect the stochastic evolution
of xt and st under the physical measure P. Finally, the ergodic density of (xt, st) under
P is constructed as follows. First, I find the unitary eigen-vector ph(xk, s)0≤k≤Nh,1≤s≤Ns
(associated with the eigen-value 1) of the transpose of a Markov matrix whose elements
correspond to transition probabilities in and out of performing states (xk, s) (for xk ≤ xs),
as well as in and out of default states (xk, s) (for xk ≥ mins xs). Once again those transition
26For most parameter configurations of interest, the issuance policy is increasing in the debt price schedule,and the debt price is a decreasing function of the issuance schedule. Thus, without dampening, the algorithmends up frequently in an infinite loop: a high debt price at the end of iteration j leads to a high issuancepolicy in iteration j + 1; such high issuance policy feeds back into a low debt price at iteration j + 1, whichleads to a low issuance policy in iteration j + 2, thus creating the infinite loop.
102
probabilities are constructed in an identical way to those described previously. For each state
(xk, s), the ergodic density at such point is approximated by πh(xk, s), computed as follows:
πh(xk, s) :=ph(xk, s)∆t
hP(xk, s)∑
xj ,s′ph(xj, s′)∆thP(xj, s′)
In order to compute CDS prices, I use a slightly modified procedure. As described previ-
ously, I need to compute the risk-neutral expected loss Ls(x, T ) and the risk-neutral expected
present value of CDS premia Ps(x, T ). Introduce the constant ε > 0. QhQ(x, s) and ∆thQ(x, s)
are now defined as follows:
QhQ(x, s) := x2|σs|2 + h|µQ(x, s)|+ h/ε
∆thQ(x, s) :=h2
QhQ(x, s)
I still have infx,sQhQ(x, s) > 0, which means that ∆thQ(x, s) is well defined. For all x, s, I have:
limh→0
∆thQ(x, s) = 0
The state space now includes time-to-maturity T , and the approximating Markov chain is
now the three-dimentional process XhQ,n :=
(xhQ,n, s
hQ,n, T
hQ,n). Given a starting state Xh
Q,n =
(x, s, T ), I define the following transition probabilities:
Pr(Xh
Q,n+1 = (x+ h, s, T )|XhQ,n)
=eΛss∆thQ(x,s)
QhQ(x, s)
(x2|σs|2
2+ hmax (0, µQ(x, s))
)Pr(Xh
Q,n+1 = (x− h, s, T )|XhQ,n)
=eΛss∆thQ(x,s)
QhQ(x, s)
(x2|σs|2
2+ hmax (0,−µQ(x, s))
)Pr(Xh
Q,n+1 = (x, s, T − εh)|XhQ,n)
=eΛss∆thQ(x,s)h/ε
QhQ(x, s)
Pr(Xh
Q,n+1 = (x, s′, T ) |XhQ,n)
=
(Λss′
−Λss
)(1− eΛss∆thQ(x,s)
)Notice that these transition probabilities are all greater than zero, less than 1, and they add
up to 1. The Markov chain created satisfies local consistency conditions similar to those in
equations (69), (70), (71), (72) and (73), in addition to:
Es,x[∆T hQ,n
]= −∆thQ(x, s)
vars,x[∆T hQ,n
]= o
(∆thQ(x, s)
)103
The state space gridGh is three-dimensional, and is of the form (xi, s, Tj)1≤i≤Nx,h,1≤s≤Ns,1≤j≤NT,h ,
where the grid points xi and the grid points Tj are equally spaced with distance h and
εh respectively. For xs > xi ≥ 0 and for T ≥ Ti > 0, I compute Ls(xi, Tj) and Ps(xi, Tj) on
the grid Gh as follows:
Ls(xi, Tj) = e−rs∆thQ(xi,s)
∑(x′Q,s′Q,T ′Q)
Pr(X ′Q|(xi, s, Tj)
)Ls′Q
(x′Q, T
′Q)
Ps(xi, Tj) = ∆thQ(xi, s) + e−rs∆thQ(xi,s)
∑(x′Q,s′Q,T ′Q)
Pr(X ′Q|(xi, s, Tj)
)Ps′Q
(x′Q, T
′Q)
I note xNx,h,s the grid point at which the sovereign government defaults optimally in state s, in
other words Nx,h,s = xs/h. The boundary conditions at T = 0 are Ls(xi, 0) = 0 for i < Nx,h,s,
Ls(xi, 0) = (1−D(xi)) for i ≥ Nx,h,s, and Ps(xi, 0) = 0 for all xi. The boundary conditions at
T > 0 are Ls(xi, T ) = (1−D(xi)) and Ps(xi, T ) = 0 for i ≥ Nx,h,s and for any T . This system
of linear equations is solved recursively: starting from Ls(xi, 0), Ps(xi, 0)1≤i≤Nx,h,s≤Ns , I can
compute Ls(xi, εh), P (xi, εh)1≤i≤Nx,h,s≤Ns via the system of linear equations above, and
progress backwards.
104
A.4 Approximating Continuous State Markov Processes
Several methods have been implemented over the years to approximate continuous state
Markov processes by discrete state Markov chains. Some of these methods are described in
Tauchen (1986), Tauchen and Hussey (1991), and more recently Tanaka and Toda (2013)
and Tanaka and Toda (2015), when dealing with highly persistent processes.
The methods mentioned above work well with a relatively large number of states. Since
I use a small number of discrete states in my numerical application, I will use an ad-hoc
procedure relying on matching conditional and unconditional moments of the original process
and my approximating process. I use the stochastic discount factor of Lettau and Wachter
(2007) to illustrate my procedure. In their article, the pricing kernel features (a) a constant
risk-free rate, and (b) a time-varying price of risk νt that follows an AR(1) process – the
discrete time equivalent of an Ornstein-Uhlenbeck process:
dνt = −κν (νt − ν) dt+ σνdZt
Lettau and Wachter (2007) parameterize ν, κν , σν according to Table 6.
Table 6: Parameters for Lettau and Wachter (2007) Model
Parameter Variable Value (annualized)Average risk-price ν 62.5%